text:
Order gift certificat
The responsible party in the sense of the data protection laws is:
Art on Ice Production AG, Rotfluhstrasse 91, 8702 Zollikon.
When you access our website, information of a general nature is automatically collected. This information (server log files) includes, for example, the type of web browser, the operating system used, the domain name of your Internet service provider and the like. This is exclusively information that does not allow any conclusions to be drawn about your person. This information is technically necessary in order to correctly deliver the content of web pages requested by you and is mandatory when using the Internet. Anonymous information of this kind is statistically evaluated by us in order to optimize our Internet presence and the technology behind it.
Like many other websites, we also use so-called "cookies". Cookies are small text files that are transferred from a website server to your hard drive. Through this, we automatically receive certain data such as IP address, browser used, operating system about your computer and your connection to the Internet.
Cookies cannot be used to run programs or deliver viruses to a computer. Based on the information contained in cookies, we can facilitate your navigation and enable the correct display of our web pages.
Under no circumstances will the data we collect be passed on to third parties or linked to personal data without your consent.
Of course, you can also view our website without cookies. Internet browsers are regularly set to accept cookies. You can deactivate the use of cookies at any time via your browser settings. Please use the help functions of your Internet browser to find out how to change these settings. Please note that individual functions of our website may not work if you have disabled the use of cookies.
When registering to use our personalized services, some personal data is collected, such as name, address, contact and communication data such as telephone number and e-mail address. If you are registered with us, you can access content and services that we offer only to registered users. Registered users also have the option, if necessary, to change or delete the data provided during registration at any time. Of course, we will also provide you with information about the personal data we have stored about you at any time. We will also be happy to correct or delete this data at your request, provided that there are no legal obligations to retain it. To contact us in this context, please use the contact details provided at the end of this privacy policy.
In order to provide chargeable services, we request additional data, such as payment details.
To protect the security of your data during transmission, we use state-of-the-art encryption methods (e.g. SSL) via HTTPS.
When registering to receive our newsletter, the data you provide will be used exclusively for this purpose. Subscribers may also be informed by e-mail about circumstances relevant to the service or registration (for example, changes to the newsletter offer or technical circumstances).
For an effective registration we need a valid e-mail address. The data will be used exclusively for the newsletter dispatch and will not be passed on to third parties.
You can revoke your consent to the storage of your personal data and its use for newsletter dispatch at any time. You will find a corresponding link in each newsletter. In addition, you can unsubscribe at any time directly on this website or inform us of your wish to do so via the contact option provided at the end of this privacy policy.
If you contact us by e-mail or contact form, the information you provide will be stored for the purpose of processing the request and for possible follow-up questions.
This website uses Google Analytics, a web analytics service provided by Google Inc (hereinafter: Google). Google Analytics uses "cookies", which are text files placed on your computer, to help the website analyze how users use the site. The information generated by the cookie about your use of this website is usually transmitted to a Google server in the USA and stored there. However, due to the activation of IP anonymization on these web pages, your IP address will be truncated beforehand by Google within member states of the European Union or in other contracting states of the Agreement on the European Economic Area. Only in exceptional cases will the full IP address be transmitted to a Google server in the USA and shortened there. On behalf of the operator of this website, Google will use this information for the purpose of evaluating your use of the website, compiling reports on website activity and providing other services relating to website activity and internet usage to the website operator. The IP address transmitted by your browser as part of Google Analytics will not be merged with other data from Google.
You may refuse the use of cookies by selecting the appropriate settings on your browser, however please note that if you do this you may not be able to use the full functionality of this website. You can also prevent the collection of data generated by the cookie and related to your use of the website (including your IP address) to Google and the processing of this data by Google by downloading and installing the browser plugin available at the following link: Browser add-on to disable Google Analytics.
For more information on terms of use and data protection, please visit http://www.google.com/analytics/terms/de.html or https://www.google.de/intl/de/policies/.
This website uses the online advertising program "Google AdWords" and as part of it the conversion tracking. Google AdWords sets a cookie on your computer if you have reached our website via a Google ad. These cookies lose their validity after 30 days and are not used for personal identification. If the user visits certain pages of our website and the cookie has not yet expired, we and Google can recognize that the user clicked on the ad and was redirected to this page. Each Google AdWords customer receives a different cookie. Cookies can therefore not be tracked through the websites of AdWords customers.
The information obtained using the conversion cookie is used to create conversion statistics for AdWords customers who have opted for conversion tracking. Customers learn the total number of users who clicked on their ad and were redirected to a page tagged with a conversion tracking tag. However, they do not receive information that personally identifies users.
If you do not wish to participate in the tracking process, you can easily deactivate the Google conversion tracking cookie via your internet browser under user settings. You will then not be included in the conversion tracking statistics. You can find out more about this via Google's privacy policy at http://www.google.de/policies/privacy/.
We integrate the maps of the service "Google Maps" of the provider Google LLC, 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA. The processed data may include, in particular, IP addresses and location data of the users, which, however, are not collected without their consent (usually executed as part of the settings of their mobile devices). The data may be processed in the USA.
Privacy policy: https://www.google.com/policies/privacy/
We embed YouTube videos on some of our websites. The operator of the corresponding plugins is YouTube, LLC, 901 Cherry Ave, San Bruno, CA 94066, USA. When you visit a page with the YouTube plugin, a connection to YouTube servers is established. This tells Youtube which pages you are visiting. If you are logged into your Youtube account, Youtube can assign your surfing behavior to you personally. You can prevent this by logging out of your Youtube account beforehand.
If a Youtube video is started, the provider uses cookies that collect information about user behavior.
If you have deactivated the storage of cookies for the Google Ad program, you will not have to deal with such cookies when watching Youtube videos. However, Youtube also stores non-personal usage information in other cookies. If you would like to prevent this, you must block the storage of cookies in the browser.
Further information on data protection at "Youtube" can be found in the provider's privacy policy at: https://www.google.de/intl/de/policies/privacy/
We integrate plug-ins of the social network Facebook (address: 1601 South California Avenue, Palo Alto, CA 94304, USA) on our websites.
If you call up one of our websites with a Facebook plug-in, a direct connection is established with Facebook. We have no influence on the type and scope of the data that is collected, stored and processed by Facebook. We can only inform you according to our level of knowledge.
Facebook is informed via the plug-in that you have accessed the corresponding website of our offer. This also applies to users who are not registered with Facebook. In that case, there is the possibility that Facebook stores the IP address.
If you are a member of Facebook and logged in, the call of a website with Facebook plug-in can be clearly assigned to your user account at Facebook. In addition, the plug-ins transmit all interactions, for example when you use the Like button or leave a comment.
To prevent Facebook from storing data about you, please log out of Facebook before visiting our websites. In addition, you can block the plug-ins with add-ons for the browser….
Information on the data protection provisions, the purpose and scope of data collection and data processing as well as the setting options for protecting your privacy at Facebook can be found here: http://www.facebook.com/privacy/explanation.php.
This offer uses functions of the Twitter service. This is provided by Twitter Inc, 1355 Market St, Suite 900, San Francisco, CA 94103, USA. Twitter offers the so-called "Tweet" function. If you use the Twitter function on our websites, the websites you visit will be linked to your Twitter account and possibly made known to other users. In this process, data is also transmitted to Twitter.
As a provider, we do not receive any knowledge of the content of the transmitted data and its use by Twitter. For more information, please refer to Twitter's privacy policy at http://twitter.com/privacy.Twitter. The following link allows you to define your own privacy settings: http://twitter.com/account/settings.
Our website uses functions of the service Instagram (Instagram Inc., 1601 Willow Road, Menlo Park, CA, 94025, USA). This may include, for example, content such as images, videos or texts and buttons with which users can share content of this online offer within Instagram. If the users are members of the Instagram platform, Instagram can assign the call of the content and functions to the profiles of the users there. Privacy policy of Instagram: http://instagram.com/about/legal/privacy/.
In order to display our content correctly and graphically appealing across browsers, we use script libraries and font libraries such as web fonts from fonts.com on this website. Web fonts are transferred to your browser's cache to avoid multiple loads. If the browser does not support Google web fonts or prevents access, content is displayed in a standard font.
Calling up script libraries or font libraries automatically triggers a connection to the operator of the library. In the process, it is theoretically possible that operators of corresponding libraries collect data.
The privacy policy of the library operator fonts.com can be found here: https://www.fonts.com/info/legal.
Right to information, correction, blocking, deletion and objection
You have the right to obtain information about your personal data stored by us at any time. You also have the right to have your personal data corrected, blocked or, apart from the mandatory data storage for business processing, deleted. Please contact our data protection officer for this purpose. You will find the contact details at the very bottom.
To ensure that a blocking of data can be taken into account at any time, this data must be kept in a blocking file for control purposes. You can also request that the data be deleted, unless there is a legal archiving obligation. If such an obligation exists, we will block your data upon request.
You can make changes or revoke consent by notifying us accordingly with effect for the future.
Change privacy policy
We reserve the right to occasionally adapt this data protection declaration so that it always complies with the current legal requirements or in order to implement changes to our services in the data protection declaration, e.g. when introducing new services. The new privacy policy will then apply to your next visit.
If you have any questions regarding data protection, please send us an e-mail or contact our data protection officer directly:
Get the latest info and offers.
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text:
Listed by District Number
Listed by Committee Assignments
Listed by Standing Committees
Who's My Legislator?
Right-to-Know Law
Legislation Quick Search
Pennsylvania State Senate
https://www.legis.state.pa.us/cfdocs/legis/BiosHistory/MemBio.cfm?ID=5233&body=S
Home / Historical Biographies / Andrew G Miller
Andrew G Miller
Session Year
1869 18 Democrat
Cumberland, York
Father of Sen. William E. Miller, Andrew G. was born on June 7, 1811, Yellow Breeches Creek, Cumberland County, the son of Abraham and Elizabeth (Boyer) Miller. A fuller, he married Eleanor Umberger and built a fulling mill on Conodaguinet Creek, West Pennsboro Township. He later moved the family to West Hill, West Pennsboro Township, opening a general store, hotel, and blacksmith shop in Newville. He eventually opened another store in Dickinson Township. After he incurred an arm injury at the blacksmith shop, resulting in the limb's amputation, he sold his business interests and retired to farming. He organized, served as cashier, and later became president of the Farmers and Mechanics Bank of Shippensburg. He died on February 14, 1880 in Shippensburg.
Interested in visiting the State Capitol? For information on visiting the Capitol Complex, please visit pacapitol.com
Contact Senator
#PaSenate MAJORITY POLICY COMMITTEE meets today, 7/15/2019 at 10:00 AM: yesterday at 6:17 AM
#PaSenate LOCAL GOVERNMENT meets today, 7/11/2019 at 10:00 AM: http://tinyurl.com/y68f4run about 5 days ago
#PaSenate Committee Meeting, MAJORITY POLICY COMMITTEE, 07/15/2019, 10:00 AM: http://tinyurl.com/y28b26gc#07/15/2019 about 6 days ago
Follow @PaLegis
Right-to-Know Law/ Records Request
Senate Library
Special Access for the Hearing/Speech Impaired

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: 'SELECT AddGeometryColumn' in C++ crashed I am using MSVC 2018 developed a GIS program.  I use spatialite API to create a SQLite geometry database.  The program always crashed when executing "Select AddGeometryColumn'. I don't know if I need to include a preprocessor name or miss a library in my project file.  I included the following libraries,
spatialite_i.lib
sqlite3_i.lib
proj_5_2.lib
lwgeom_i.lib
freexl_i.lib
geos_c.lib
iconv.lib

My program as the following,
CString sSRID = sSRID.Format("%d", srid);
CString qrybuf = CString("") + "SELECT AddGeometryColumn('" + recs->code + "','geom', " + sSRID + ", 'POINT', 'XY');";
char *errmsg;
int result = sqlite3_exec(sqliteDS->getHandle(), qrybuf.GetString(), NULL, NULL, &errmsg);
if (result != SQLITE_OK) {
    acad_printf("failed to add geometry column. error: %s\n", errmsg);
    sqlite3_free(errmsg);
    return;
}

Can you advise?

redpajama_set_name: RedPajamaStackExchange


Every Sunday morning we provide a safe, fun and creative environment for kids from birth through sixth grade where they can learn about Jesus and connect with each other. Our volunteers are the heartbeat of Relate Church, and it's an honor to serve with each and every one of them as they love on kids each week! Every volunteer who leads in Relate Kids, has had a background check and is committed to making the ministry fun and Biblically relevant, while meeting the health and safety needs of your child.
Our Goal is to create a peaceful atmosphere and care for each Relate Baby. Their physical needs and first impression of our Heavenly Father is our main focus. During service, your child will have plenty interaction, playtime and a lession built around their developmental needs. We love each Relate baby like they are our own!
YOU CAN PRE-REGISTER YOUR CHILD BEFORE YOUR VISIT.
Jrs. are special little people! Preschoolers learn best through actions and repetition. That is why we structure their time with us around appropriate activities, using creative and fun Bible-based curriculum. We want to help you cultivate in them a strong foundation of trust and faith in God. By meeting the health and safety needs of your child and by giving you the security of knowing your little one is being cared for and taught by loving teachers in a safe and secure environment, we are confident that they will grow leaps and bounds!
During services, your child will be in the care of trained volunteers all of whom meet required background standards.
We believe that when you are confident in leaving your baby in the care of our staff, then you are able to worship with a sense of security. With great care, we have prepared our nursery rooms to be the best environment for your baby with age appropriate toys and an opportunity to interact with others. As we hold the babies, we pray for them to receive all that God has for them in life. We believe every child we pray for will experience the love of God.
Our goal is to encourage your little one in a growing, loving relationship with Jesus Christ and others in an atmosphere where they feel a sense of belonging. We present biblical learning materials with a hands-on approach to help establish in the hearts of the children an active faith toward God and His Word. We teach the simplicity of the Gospel through a variety of stories, crafts, Bible action songs and other activities.
Every lesson prepared and taught reinforces Bible truths to your preschooler. Through fun, interactive bible stories, we share the love of Jesus and the hope of the gospel. Our hope is to encourage your JR. as they grow in their relationship with Jesus.
Relate Kids is for children ages Kindergarten to sixth grade. From the time a child is born until they graduate High School, you'll have 940 weekends. If your child is in Kindergarten you have 637 left. Life happens fast. At Relate we desire to use these remaining weekends to partner with you to help your child discover and embrace a growing relationship with Jesus. Each week at Relate Kids your child will experience an exciting, fun and character-building time in our children's services designed just for them. We look forward to seeing you and your children soon at Relate Church.
Fifth and Sixth graders are anything but stuck in the middle at Relate. We believe that they are leaders to the elementary kids and we want them to continue to be leaders when they enter Relate Students. 56 is a extra gathering for kids in 5th and 6th grade on the second Wednesday of every month. They will have worship, fun, fellowship and time in the Word.


We offer returns for most items within 30 days of receipt of shipment. To initiate a return, please email boy@noiseappeal.com and make sure to include the order number.
• CDs, vinyl records, cassettes, and DVDs must be unopened and still in their plastic wrap.
• Clothing must be unwashed and unworn.
In the unfortunate case that your items are damaged or defective upon delivery, please boy@noiseappeal.com to discuss replacement or refund options.
Please include the item name and order number in your email.
Due to the nature of digital goods, they are not refundable. If you are experiencing problems downloading your digital goods, please contact boy@noiseappeal.com.


This shoe has been designed to practise sport even in drastic weather conditions. The use of technical warm materials, such as the special heat windtex lining and the outward waterproof lycra covering, keep the foot warm and dry.
The new fastening system is easy and quick because of the waterproof anti-slide zipper. The shoe upper has been injected with a special rubber and the use of technical devices assures a very good resistance against any excoriations, debris or impacts.
Suited to very light MTB, this shoe really looks like CR2. We have made it stronger by inserting on its top an anti-tear material and putting on the b..
DescriptionSuited to both professionals and amateurs who like simplicity and comfort during each session. Very special design but not too overly, made..
This shoe has been designed to practise sport even in drastic weather conditions. The use of technical warm materials, such as the special heat windte..
Descriptioncrono-team-CR4Suited to both professionals and amateurs who like the practical three strap shoe, CR4 has a simple design but it is smartly ..
DescriptionThe performance of this model, which is realized by means of a very high light and perspiring microfiber, has been further improved by the ..


text:
THE LION KING TICKETS
THE LION KING theatre Breaks
Musical Lion king Show,Tickets at Lyceum Theatre
Lyceum Theatre London having been present in London since 1765; could be regarded as the oldest theatre in London. It is a West End theatre, located on Wellington Street, off the Strand, in the City of Westminster, London. It was notable architect Samuel Beazley who designed the current structure that opened on 14th July 1834. In 1904, the theatre was rebuilt by Bertie Crewe which he decorated in rich rococo style but retained the facade and portico designed by Beazley.
In 1809, the theatre evolved as a license house and was used by the Drury Lane Company for hosting dramatic performance until 1812 after their theatre was burnt down in fire. In 1816, Samuel Arnold rebuilt the house and was named The English Opera House when it reopened. The Lyceum Theatre became famous for being the first theatre in London to be lit by gas, for hosting performances of Mozart's Cosi fan tutte and for being the home to Lion King.
Disney's animated classic Musical Lion King Show has been running at the Lyceum Theatre London since 1999. The musical has scores from the legendry composer Elton John and lyrics by Tim Rice. Lyceum Theatre Tickets London are best to enjoy the Lion King on any weekday or on weekends. The piece is suitable for audiences of every genre and is great when watched in the Company of close ones.
21 Wellington Street
WC2E 7RQ
2 hours 30 minutes. Tuesday - Saturday at 7.30pm, Wednesday, Saturday & Sunday 2.30pm
Booking until 05 January 2014 ------------------------------------------
Call us now on + 44 (0) 870 1403 300 to make a Lion King group booking! Disney's The Lion King. Please note there are now no school group rates available until September 2013.
Performance times: Tuesday - Saturday at 7.30pm,Wednesday, Saturday and Sunday 2.30pm
There are no school group rates until 1st September. In June and July there are some regular group rates on certain Fridays at a higher rate than the below, subject to availability. Please call + 44 (0) 870 1403 300 or email for information.
Exclusions: There are no group or school rates 27 to 31 May, 25 October to 3 November, 20 December to 05 January 2014. Please also note that group rates are no longer valid Tuesday to Thursday until 1 September
Booking until 05 January 2014
Children under 3 years of age will not be admitted to performances of Disney's The Lion King at the Lyceum Theatre and all children under the age of 16 must be accompanied by an adult. Strobe lighting is used several times during the performance. Please note, we sell tickets for the LONDON production of the show.
Lion King the Musical Tickets I Lyceum Theatre London I Group Booking I Lyceum Theatre I Lion King London I Sitemap I
Theatre Lyceum I Lion King Show I Lion King Theatre I The Lion King
lyceumtheatrelondon.com is not affiliated with any Official Owners of the Theatre or Organizers. All tickets sold are official tickets and authentic.
Every effort has been made to ensure the accuracy of the information contained in these pages. We accept no responsibility for any errors.
Copyright © 2012 London Westend Theatre. All rights reserved.


The coolest little Surf Lodge in Newquay, Cornwall, United Kingdom.
Ocean Lodge is just minutes from the world-famous Fistral surf beach, bars, restaurants and local attractions like Newquay Zoo, Newquay Aquarium – or simply use the hotel as a base to explore stunning Cornwall!
The ideal cheap & cheerful budget hotel accommodation for large groups, Stags, Hens, Sports or University Tours and Schools.
Rooms are very well maintained with cool, modern décor and comprise mainly of bunk accommodation with some single or double beds. Exclusive booking of the hotel can be arranged for large groups.
Continental Breakfast included in price (hot lunch, hot meals available upon request).
This property is located near Tolcarne Wedge. A busy and popular body-boarding wave, this spot works best on a high tide. A clean and powerful wave, best in winter. Big tidal changes result in a massive beach to explore at low tide.
This property is also located near Towan. This sandy bottomed beach break is prone to closing out, but when a winter swell hits it can be one of the best options around, and can hold some size. When the swell is big enough a nice left will break near the harbor. The wave will work on all tides but low tide producing bigger waves. Towan is very popular with everyone from groms to local rippers, be ready for a crowd.
This property is also located near Fistral Beach. On a good day, one of England's best beach breaks. Waves can be hollow and rides can be long. Summer weekend crowds can be intense. There are multiple sections to surf at the expansive sandy beach bay each with it's own personality and tidal temperament. Catch this spot on a sunny day with light offshores and a good swell and you'll be in paradise!


Easy to read and accurate water level indicator.
Monitoring temperature by thermometer. – Capable of brewing different types of tea.
Screw dial counting potentiometer for setting desired temperature and time.
Heat Insulator keeps the water/tea warm.
The circle band light works by device operation.
Dimensions: 9.5″ L, 7.9″ W, 8.7″ H.
Utensil were design to make eating easier.


Meditation in motion-enjoy the sled dog ride sitting on the sled in the magic arctic night. We hope to see the northern lights.
Husky Dream- drive your own team, moose safari and at home visit!
Experience the arctic wilderness with your own team of sled dogs.
On this invigorating lunch dog sledding trip you will travel as a passenger on a sled driven by one of our professional guides.
Kiruna offers a wide range of activities from snowmobiling to in the winter through to hiking in the summer. Be sure not to miss venturing out into the wilderness by dog sled on a crisp winter's day a highlight of many guest to the region.
Whilst those visiting in the Autumn can explore the many hiking trails and wonderful colours of the Autumn leaves, Kiruna Northern Lights experiences are popular as it's very well positioned to see the spectacular Northern Lights any evening from mid September through to late March.
Kiruna Northern Lights trips are very popular, especially when paired with other activities such as Kiruna dog sledding. Plan your perfect holiday itinerary in advance to avoid missing out.
Kiruna or Giron is the northernmost town of Sweden sitting at the meeting point of great forests and the massive mountains that sit between Sweden and Norway. Kiruna is a great place to visit offering year round Arctic activities including dog sledding and Northern Lights in the winter and white water rafting in the summer. The world famous ICEHOTEL and ICEHOTEL365 are also located just a short drive from the city.


text:
/*
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package com.facebook.presto.common.block;

import org.openjdk.jol.info.ClassLayout;
import org.testng.annotations.Test;

import static com.facebook.presto.common.type.BigintType.BIGINT;
import static org.testng.Assert.assertEquals;
import static org.testng.Assert.assertTrue;

public class TestArrayBlockBuilder
{
    // ArrayBlockBuilder: isNull, offset, 3 * value (FixedWidthBlockBuilder: isNull, value)
    private static final int THREE_INTS_ENTRY_SIZE = Byte.BYTES + Integer.BYTES + 3 * (Byte.BYTES + Long.BYTES);
    private static final int EXPECTED_ENTRY_COUNT = 100;

    @Test
    public void testArrayBlockIsFull()
    {
        testIsFull(new PageBuilderStatus(THREE_INTS_ENTRY_SIZE * EXPECTED_ENTRY_COUNT));
    }

    private void testIsFull(PageBuilderStatus pageBuilderStatus)
    {
        BlockBuilder blockBuilder = new ArrayBlockBuilder(BIGINT, pageBuilderStatus.createBlockBuilderStatus(), EXPECTED_ENTRY_COUNT);
        assertTrue(pageBuilderStatus.isEmpty());
        while (!pageBuilderStatus.isFull()) {
            BlockBuilder elementBuilder = blockBuilder.beginBlockEntry();
            BIGINT.writeLong(elementBuilder, 12);
            elementBuilder.appendNull();
            BIGINT.writeLong(elementBuilder, 34);
            blockBuilder.closeEntry();
        }
        assertEquals(blockBuilder.getPositionCount(), EXPECTED_ENTRY_COUNT);
        assertEquals(pageBuilderStatus.isFull(), true);
    }

    //TODO we should systematically test Block::getRetainedSizeInBytes()
    @Test
    public void testRetainedSizeInBytes()
    {
        int expectedEntries = 1000;
        BlockBuilder arrayBlockBuilder = new ArrayBlockBuilder(BIGINT, null, expectedEntries);
        long initialRetainedSize = arrayBlockBuilder.getRetainedSizeInBytes();
        for (int i = 0; i < expectedEntries; i++) {
            BlockBuilder arrayElementBuilder = arrayBlockBuilder.beginBlockEntry();
            BIGINT.writeLong(arrayElementBuilder, i);
            arrayBlockBuilder.closeEntry();
        }
        assertTrue(arrayBlockBuilder.getRetainedSizeInBytes() >= (expectedEntries * Long.BYTES + ClassLayout.parseClass(LongArrayBlockBuilder.class).instanceSize() + initialRetainedSize));
    }

    @Test(expectedExceptions = IllegalStateException.class, expectedExceptionsMessageRegExp = "Expected current entry to be closed but was opened")
    public void testConcurrentWriting()
    {
        BlockBuilder blockBuilder = new ArrayBlockBuilder(BIGINT, null, EXPECTED_ENTRY_COUNT);
        BlockBuilder elementBlockWriter = blockBuilder.beginBlockEntry();
        elementBlockWriter.writeLong(45).closeEntry();
        blockBuilder.appendStructure(new LongArrayBlockBuilder(null, 1).writeLong(123).closeEntry().build());
    }
}

redpajama_set_name: RedPajamaGithub


Puma Biotechnology, Inc. (Nasdaq: PBYI), a biopharmaceutical company, and Specialised Therapeutics Asia, an international biopharmaceutical company with strategic focus and expertise in Australia, New Zealand and South East Asia, have entered into an exclusive agreement under which Specialised Therapeutics will commercialize NERLYNX® (neratinib) throughout South East Asia, beginning with Australia, Singapore, Malaysia, Brunei and New Zealand. Currently, Specialised Therapeutics markets Abraxane® and other oncology products in these countries.
NERLYNX is not approved currently for commercialization outside of the United States. Specialised Therapeutics will be responsible for seeking the requisite regulatory approvals and, once approved, for commercializing NERLYNX in those countries. Puma will receive upfront and milestone payments of up to $4.5 million throughout the term of this agreement, as well as significant double digit royalties on NERLYNX sales in all regions in which Specialised Therapeutics commercializes NERLYNX.
Puma Biotechnology, Inc. is a biopharmaceutical company with a focus on the development and commercialization of innovative products to enhance cancer care. The Company in-licenses the global development and commercialization rights to three drug candidates — PB272 (neratinib (oral)), PB272 (neratinib (intravenous)) and PB357. NERLYNX® (neratinib) is approved for commercial use by prescription in the United States as extended adjuvant therapy for early stage HER2-positive breast cancer following adjuvant trastuzumab-based therapy and is marketed as NERLYNX. Neratinib is a potent irreversible tyrosine kinase inhibitor that blocks signal transduction through the epidermal growth factor receptors, HER1, HER2 and HER4. Currently, the Company is primarily focused on the commercialization of NERLYNX and the continued development of its other advanced drug candidates directed at the treatment of HER2-positive breast cancer. The Company believes that NERLYNX has clinical application in the potential treatment of several other cancers that over-express or have a mutation in HER2.
Specialised Therapeutics Asia Pte Ltd (ST Asia) is an international biopharmaceutical company established to provide pioneering healthcare solutions to patients throughout South East Asia, including Australia and New Zealand.
ST Asia and its regional affiliates collaborate with leading global pharmaceutical and diagnostic companies to bring novel, innovative and life changing healthcare solutions to patients affected by a range of diseases. ST Asia is committed to making new and novel therapies available to patients around the world, targeting diseases where there remains an unmet medical need. ST Asia's broad therapeutic portfolio currently includes novel agents in oncology, haematology, neurology, ophthalmology and supportive care.
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text:
Stuart Deutsch
University Professor and Willard Heckel Scholar
sdeutsch@law.rutgers.edu
Stuart L. Deutsch served as dean of Rutgers Law School-Newark from 1999-2009 and Acting Senior Associate Dean from 2014-2016. He specializes in property, environmental law, and urban development and has served on development and has served on committees for the New Jersey Supreme Court, New Jersey Institute of Continuing Legal Education and the American Bar Association.
Stuart L. Deutsch served as dean of Rutgers School of Law–Newark from July 1999 until July 2009, when he stepped down and was awarded the title University Professor. In 2014, he was named Acting Senior Associate Dean, which he held until June 30, 2016, and Willard Heckel Scholar. As Acting Senior Associate Dean, he was involved directly with admissions, financial aid, academic support, career services, the minority student program, pro bono and public interest programs, and student services overall. From 2012-2016, he also was Director of Global and International Programs for the law school.
In this position, he is working to develop a range of international and comparative law courses in the academic curriculum; study-abroad, pro bono, and professional work opportunities for students in foreign countries; international exchange and research opportunities for faculty; interdisciplinary study opportunities with other graduate units of the university; and a flow of international speakers and guests at the law school.
He received his J.D. from Yale Law School in 1969 and his LL.M. from Harvard Law School in 1974, where he was a Fellow in Law and the Humanities. He is a 1966 graduate of the University of Michigan.
Professor Deutsch has been deeply involved with issues relating to the legal profession, ethics, diversity, and legal education, including membership on the New Jersey Supreme Court's Commission on Professionalism; the Board of the New Jersey Institute for Continuing Legal Education; and the American Bar Association's Section of Legal Education and Admissions to the Bar Questionnaire Committee. He also has worked with the ABA Section of Legal Education and Admissions to the Bar Technology Committee; the ABA's Central and Eastern European Law Initiative; the Association of American Law Schools' Membership Committee; the World Conservation Union's Commission on Environmental Law; and advisory committees to reform the housing and eviction courts. He also has been chair of the Environmental Law and Local Government Law Sections of the Association of American Law Schools.
Professor Deutsch has served on the board of directors of several environmental organizations, as a hearing officer for the Chicago Commission on Human Relations, and as chair of fair housing and housing development organizations. He was a pro bono attorney for the Leadership Council for Metropolitan Open Communities, the umbrella fair housing organization in the Chicago metropolitan area.
Formerly a professor of law, co-director of the Program in Environmental and Energy Law, and founding director of the Institute for Science, Law and Technology at the Chicago-Kent College of Law at the Illinois Institute of Technology, he also served as interim dean in 1996-97, and as associate dean for academic affairs and associate dean for interdisciplinary programs. Professor Deutsch's expertise is in the fields of property, environmental law, and urban development.
He is the author of Deutsch's Illinois Environmental Laws Annotated and for 18 years was co-editor of Land Use and Environment Law Review. Professor Deutsch also has been a visiting faculty member at the University of Illinois College of Law; an associate professor at the University of Santa Clara School of Law, where he began his teaching career; and an associate with a law firm in New York City.
CRIT LEGAL ANALYSIS
LOW INC HOUSING SEM


NICE today recommends in draft guidance that immunotherapy drug pembrolizumab (also known as Keytruda), used with pemetrexed and platinum chemotherapy, should be a treatment option for people with untreated advanced lung cancer.
NICE estimates that more than 3,000 patients in England may be eligible for this combination treatment.
Made by Merck, Sharpe & Dohme (MSD), pembrolizumab works by targeting a specific protein on the surface of cells which is involved in the body's immune response to cancer.
MSD reported that in a clinical trial, this pembrolizumab combination therapy gave patients nearly 4 additional months without their cancer progressing, compared with standard treatment of pemetrexed with platinum chemotherapy.
NICE said that as further evidence is needed on the benefits of the treatment, it could not be considered for routine NHS use.
However, it recognised that it has the potential to be cost effective, and was likely to extend life. This made it eligible for use on the Cancer Drugs Fund.
Under the CDF managed access agreement MSD will continue to collect data on the treatment's effectiveness until June 2019, after which there will be a further review of this guidance. The treatment, which individual patients can receive for a maximum of two years, will continue to be available during this review.
Meindert Boysen, director of the NICE Centre for Health Technology Evaluation, said: "Over the past two decades there have been few improvements for people with this type of lung cancer so we are pleased to support access to this new option via the Cancer Drugs Fund, and keen to see further data on its effectiveness as it becomes available.
"NICE recommends that this treatment is given for a maximum of two years, because it is thought that patients do not need to continue treatment beyond this point to achieve the benefits. NICE will consider this point when we review the additional evidence."
Professor Charles Swanton, Cancer Research UK's chief clinician, said: "This decision is good news for people affected by this type of lung cancer. This drug combination has shown significant benefits for some patients in clinical trials, helping them to survive their cancer for longer.
"This also shows the value of the Cancer Drugs Fund, which gives NICE the option to approve promising treatments while more data is collected on their long-term benefits. Lung cancer is the biggest cancer killer in the UK so it's vital that innovations like this reach patients as quickly as possible."
Pembrolizumab (brand name Keytruda) is administered every 3 weeks by intravenous infusion. The regular price of a 200mg infusion is £5,260, but the company has offered the NHS a confidential discount.
Update (10 January): no appeals were received and final guidance has been published.


text:
\section{Introduction}
\subsection{The relative character}\( \)
\DimaG{Let $\bfG$ be a  reductive group\footnote{\DimaH{By a reductive group we mean a connected algebraic reductive group}} defined over $\R$}.
In this paper, we prove that a \DimaD{relative character (a.k.a. spherical character)} of a smooth admissible \Fre  representation of moderate growth of \DimaG{$\bfG(\R)$} is holonomic.
The \DimaD{relative} character is a basic notion of relative representation theory that generalizes the notion of a character of a representation.  Let us now recall the notions of spherical pair, \DimaD{relative} character and holonomic distribution. \Rami{For the notion of smooth admissible \Fre  representation of moderate growth \DimaF{and its relation to Harish-Chandra modules} we refer the reader to \cite{CasGlob} or \cite[Chapter 11]{Wal}}.
\begin{defn}
\DimaG{Let $\bfH\subset \bfG$ be a (closed) algebraic subgroup defined over $\R$}. Let ${P}$ denote a minimal parabolic subgroup of \DimaG{$\bfG$ defined over $\R$} and $\bfB$ denote a Borel subgroup of $\bfG$  \DimaG{(possibly not defined over $\R$}).
The subgroup $\bfH$ is called  \emph{ spherical} if it has finitely many orbits on $\bfG/\bfB$. \DimaG{We will call $\bfH$ \emph{strongly real spherical} if it has finitely many orbits on $\bfG/\bfP$}.
\end{defn}

It is known that a pair \DimaG{$(\bfG,\bfH)$ is spherical if and only if $\bfH$ has an open orbit on $\bfG/\bfB$.}

\begin{defn}\label{def:SphChar}
Let $\bfH_1,\bfH_2\subset \bfG$ be spherical subgroups \DimaC{and let $\fh_i$ be the Lie algebras of $\bfH_i$}. Let $\chi_i$ be characters of $\DimaC{\fh_i}$. Let $\pi$ be a smooth admissible \Fre representation of moderate growth of $\bfG\DimaG{(\R)}$\DimaF{, $\pi^*$ be the continuous dual of $\pi$, and $\hat{\pi}\subset \pi^*$ be the smooth contragredient representation to $\pi$ (i.e. the only smooth admissible \Fre representation of moderate growth with the same space underlying Harish-Chandra module as $\pi^*$). }
Let
$\phi_1 \in (\pi^*)^{\DimaC{\fh}_1,\chi_1}$ and $\phi_2 \in (\hat \pi^*)^{\DimaC{\fh}_2,\chi_2}$ be equivariant functionals. Fix a Haar measure on $\bfG\DimaG{(\R})$. It gives rise to an action of the space of Schwartz functions $\Sc(\bfG\DimaG{(\R}))$ on $\pi^*$ and $\hat \pi^*$, and this action maps elements of $\pi^*$ and $\hat \pi^*$ to elements of $\hat \pi$ and $\hat{\hat \pi}=\pi$ \Rami{respectively. For the definition of the space of Schwartz functions $\Sc(\bfG\DimaG{(\R}))$ see, e.g., \cite{CasGlob,Wal,AGSc}}.

The \DimaD{relative} character $\xi_{\phi_1,\phi_2}$ of $\pi$, with respect to $\phi_1$ and $\phi_2$,  is the tempered distribution on $\bfG\DimaG{(\R})$ (i.e. a continuous functional on $\Sc(\bfG\DimaG{(\R}))$) defined by $\langle \xi_{\phi_1,\phi_2} , f \rangle= \langle \phi_1, \pi(f) \cdot \phi_2 \rangle$.
\end{defn}

\begin{defn}\label{def:SISDist}
\DimaG{Let $\bfX$ be an algebraic manifold defined over $\R$. Let $\xi\in \Sc^*(\bfX(\R))$ be a tempered distribution.
The singular support\footnote{a.k.a. characteristic variety}  $\SIS(\xi)$ of $\xi$}
is the \DimaG{ zero locus} in $T^*\bfX$ of all the symbols of (algebraic) differential operators that annihilate $\xi$. The distribution $\xi$ is called holonomic if $\dim \SIS(\xi) = \dim \bfX$.
\end{defn}

In this paper we prove the following theorem.
\begin{introthm}[See \S \ref{subsec:PfMain}]\label{thm:main}
In the situation of \Cref{def:SphChar}, the \DimaD{relative} character  $\xi_{\phi_1,\phi_2}$ is holonomic.
\end{introthm}

We prove \Cref{thm:main} using the following well-known statement.

\begin{prop}[See \S \ref{subsec:PfMain}]\label{prop:S}
Let $\g,\h_i$ be the Lie algebras of $\bfG$ and $\bfH_i$, $i=1,2$.
\DimaC{Identify $T^*\bfG$ with $\bfG\times\g^*$ and let}
 $$\bfS:=\{(g,\alpha) \in \bfG \times \g^*\, \mid\, \alpha \text{ is nilpotent, } \langle \alpha, \h_1 \rangle =0, \ \langle \alpha, Ad^*(g)(\h_2) \rangle =0\}.$$

Then $\SIS(\xi_{\phi_1,\phi_2}) \subset \bfS.$
\end{prop}

Note that the Bernstein inequality states that the dimension of the singular support of any non-zero distribution is at least the dimension of the underlying manifold. Thus  Theorem \ref{thm:main} follows from the following more precise version, which is the core of this paper.

\begin{introthm}[See \S \ref{sec:geo}]\label{thm:geo}
We have $\dim \bfS = \dim \bfG$.
\end{introthm}

\DimaC{
Let $\bfU:=\left \{ g \in \bfG \, \mid \, \bfS\cap T_g^*\bfG=\{(g,0)\}\right\}.$ Note that $\bfU$ is Zariski open since $\bfS$ is conic \DimaD{and closed}. It is easy to see that \Cref{thm:geo}  implies the following corollary.
\begin{introcor}\label{cor:geo}
 The set $\bfU$ is a Zariski open dense subset of $\bfG$.
\end{introcor}
}
This corollary is useful in view of the next proposition, which follows from \Cref{prop:S,cor:SmoothDist} below.
\begin{prop}\label{prop:ResSmooth}
The restriction $\xi_{\phi_1,\phi_2}|_{\DimaG{\bfU(\R)}}$ is an analytic function.
\end{prop}

\DimaF{
\begin{remark}
In general, $\bfS$ has irreducible components that can not lie in
$\SIS(\xi_{\phi_1,\phi_2})$ for any $\phi_1,\phi_2$. Indeed, $\SIS(\xi_{\phi_1,\phi_2})$ is coisotropic by \cite{Gab,KKS,Mal}, and thus, by Theorem \ref{thm:main}, Lagrangian. On the other hand, one can show that when \DimaG{$\bfG=\mathrm{GL}_{4,\R}$, and $\bfH_1=\bfH_2=\mathrm{GL}_{2,\R}\times \mathrm{GL}_{2,\R}$} embedded as block matrices inside $\bfG$, the variety $\bfS$ has non-isotropic (and thus non-Lagrangian) components.
\end{remark}
}

\subsection{Bounds on the dimension of the space of solutions}

Next we apply our results to representation theory. For this, we use the following theorem.

\begin{thm}[Bernstein-Kashiwara]\label{thm:DimSol}
Let \DimaG{$\bfX$ be \Rami{an algebraic manifold} defined over $\R$}. Let $$\{D_i \xi=0\}_{i=1... n}$$ be a system of linear PDE on $\DimaG{\Sc^*(\bfX(\R))}$ with algebraic coefficients. Suppose that the joint zero set of the symbols of $D_i$ in $T^*\bfX$ is $\dim \bfX$-dimensional. Then the space of solutions of this system  is finite-dimensional.
\end{thm}

It seems that this  theorem is not found in the literature in this formulation, however it has two proofs, one due to Kashiwara (see \cite{Kas,KK} for similar statements) and another due to Bernstein (unpublished).

In order to make our applications in representation theory more precise, we need an effective version of  this theorem. We prove such a version (see Theorem \ref{thm:FinSol} below) following Bernstein's approach, as it is more appropriate for effective bounds. We use this effective version to derive a relative version. Namely, we show that if the system depends on a parameter in an algebraic way, then the dimension of the space of solutions is bounded (see \S \ref{subsec:DimSolRel}  below).

This relative version allows us to deduce the following theorem.
\DimaD{
\begin{introthm}[See \S \ref{subsec:DimSolRel}]\label{thm:FinOrb
Let \DimaG{$\bfG$ be an  algebraic group defined over $\R$ and let $\bfX$ be an  algebraic $\bfG$-manifold} with finitely many orbits. Let $\fg$ be the Lie algebra of $\bfG$. Let $\cE$ be an algebraic $\bfG$-equivariant bundle on  $\bfX$.  Then, for any natural number $n\in \N$, there exists $C_n \in  \mathbb{N}$ such that for every $n$-dimensional representation $\tau$ of $\fg$ we have  $$\dim \Hom_{\fg}(\tau, \Sc^*(\bfX(\R),\cE))\leq C_{n},$$
where $\Hom_{\fg}$ denotes the space of all continuous $\fg$-equivariant maps.

\end{introthm}

\DimaG{
\begin{remark}
Note that the condition that $\bfG$ has finitely many orbits on $\bfX$ is equivalent to $\bfG(\C)$ having finitely many orbits on $\bfX(\C)$ but {\bf not} equivalent  to (and not implied by) $\bfG(\R)$ having finitely many orbits on $\bfX(\R)$.
\end{remark}
}

\subsection{Applications to representation theory}

Using \S \ref{sec:DimSol}, we give  a short proof of some results \DimaE{from} \cite{KO,KS}. Namely, we prove:
\begin{introthm}[See \S \ref{sec:Rep}]\label{thm:IntroMult} Let \DimaG{${G}$ be a reductive group defined over $\R$, $\bfH\subset \bfG$} be a Zariski closed subgroup, and $\fh$ be the Lie algebra of $\bfH$.
\begin{enumerate}[(i)]
\item \label{it:Fin} If \DimaG{$\bfH$ is a strongly} real spherical subgroup then, {for every \DimaE{irreducible smooth admissible \Fre  representation of moderate growth $\pi\in \Irr(\DimaG{\bfG(\R))}$}, and natural number $n\in \N$ there exists $C_n \in  \mathbb{N}$ such that} for every $n$-dimensional representation $\tau$ of $\fh$ we have  $$\dim \Hom_{\fh}(\pi,\tau) \leq C_n.$$
\item \label{it:Bound} If $H$ is a spherical subgroup and we consider only one-dimensional $\tau$ then the space is universally bounded, {\it i.e.}
there exists $C \in  \mathbb{N}$ such that $\dim (\pi^*)^{\fh,\chi} \leq C$ for any $\pi\in \Irr(\DimaG{\bfG(\R))})$ \Andrey{and any character $\chi$ of $\fh$}.
\end{enumerate}
\end{introthm}

\begin{introcor}\label{cor:strong}

 Let \DimaG{${G}$ be a reductive group defined over $\R$, $H\subset G$} be a Zariski closed reductive subgroup, and $\fh$ be the Lie algebra of $H$.

\begin{enumerate}[(i)]
\item \label{it:StFin} If the diagonal $\Delta \bfH$ is a \DimaG{strongly} real spherical subgroup in $\bfG\times \bfH$ then for every $\pi\in \Irr(\DimaG{\bfG(\R))})$ and $\tau \in \Irr(\DimaG{\bfH(\R))})$ we have finite multiplicities, i.e. $$\dim \Hom_{\fh}(\pi,\tau)< \infty .$$
\item \label{it:StBound}  If the diagonal $\Delta\bfH$ is a spherical subgroup in $\bfG\times \bfH$ then the multiplicities are universally bounded, i.e.,
there exists $C \in  \mathbb{N}$ such that for every $\pi\in \Irr(\DimaG{\bfG(\R))})$, $\tau \in \Irr(\DimaG{\bfH(\R))})$ we have $$\dim \Hom_{\fh}(\pi,\tau) \leq C .$$
\end{enumerate}
\end{introcor}
This corollary follows from Theorem \ref{thm:IntroMult} since $\Hom_{\fh}(\pi,\tau)$ lies in the space of $\Delta \fh$-invariant functionals on the completed tensor product $\pi \widehat{\otimes} \hat \tau\in \Irr(\DimaG{\bfG(\R))}\times \DimaG{\bfH(\R))})$ (see \cite[Corollary A.0.7 and Lemma A.0.8]{AMOT}). All symmetric pairs satisfying the conditions of the corollary were classified in \cite{KM}.

The inverse implications for Theorem \ref{thm:IntroMult}\DimaG{\eqref{it:Bound}} and Corollary \ref{cor:strong}\DimaG{\eqref{it:StBound}} are proven in \cite{KO}.

The results on multiplicities in \cite{KO,KS} are  stronger than Theorem \ref{thm:IntroMult} since they \DimaG{do not require  $H$ to be algebraic,}  and consider maps from the Harish-Chandra space of $\pi$ to $\tau$.
\DimaG{Also, in \cite{KO,KS} Theorem \ref{thm:IntroMult}\eqref{it:Fin} and Corollary \ref{cor:strong}\eqref{it:StFin} are proven in the wider generality of real spherical subgroups.}

In addition,
\cite[Theorem B]{KO} implies that if  $H\subset G$ is an algebraic spherical subgroup there exists  $C \in  \mathbb{N}$ such that $\dim \Hom_{\fh}(\pi,\tau) \leq C\dim \tau,$ for every $\pi\in \Irr(G\DimaG{(\R)})$ and every finite-dimensional continuous representation $\tau$ of $H\DimaG{(\R)}$. It is easy to modify our proof of Theorem \ref{thm:IntroMult}\eqref{it:Bound} to show the boundedness of multiplicities for any $\pi\in \Irr(G\DimaG{(\R)})$  and any $\tau$ of a fixed dimension, but the proof that the bound depends linearly on this dimension would require more work.

Our methods are different from the methods of \cite{KO}, which in turn differ from the ones of \cite{KS}, and the bounds given in the three works are probably very different.
}
\subsection{The non-Archimedean case}
 Theorem \ref{thm:geo} and Corollary \ref{cor:geo} hold over arbitrary fields of characteristic zero. They are useful also for $p$-adic local fields $F$, since the analogs of \Cref{prop:S,prop:ResSmooth} hold in this case, see \cite[Theorem A and Corollary F]{AGS}.
Namely, we have the following theorem.
\begin{thm}[\cite{AGS}]\label{thm:AGS}
Let $G$ be a reductive group defined over a non-Archimedean field $F$ of characteristic $0$ and let $\xi$ be a \DimaD{relative} character of a smooth admissible  representation with respect to two spherical subgroups $H_1,H_2\subset G$.
Let $S$ and $U$ be the sets defined in \Cref{prop:S,cor:geo}.
Then
\begin{enumerate}[(i)]
\item \label{thm:AGS:S} The wave front set of $\xi$ lies in $S\DimaG{(F)}$.

\item\label{thm:AGS:U} The restriction of $\xi$ to $U\DimaG{(F)}$  is given by a locally constant function.

\end{enumerate}
\end{thm}

\subsection{Related results}
In the group case, i.e. the case when $G=H\times H$ and $H_1=H_2=\Delta H \subset H \times H$, Theorem \ref{thm:main} essentially becomes the well-known fact that characters of admissible representations are holonomic distributions.

As we mentioned above, Theorem \ref{thm:IntroMult} was proven earlier in \cite{KO,KS} using different methods. An analog of Theorem \ref{thm:IntroMult}\eqref{it:Fin} over non-Archimedean fields is proven in \cite{Del} and \cite[Theorem 5.1.5]{SV} for many spherical pairs, including arbitrary symmetric pairs.

The group case of \Cref{cor:geo}, \Cref{prop:ResSmooth}, and \Cref{thm:AGS}\eqref{thm:AGS:U} is (the easy part of) the Harish-Chandra regularity theorem (see \cite{HCBul,HCReg}).  \Dima{Another known special case of these results is the regularity of Bessel functions, see \cite{LM,AGK,AG}.}

\subsection{Future applications}
Our proof of Theorem \ref{thm:IntroMult}\eqref{it:Bound} does not use the Casselman embedding theorem {(Theorem~\ref{thm:CasSubRep})}.
This gives us hope that it can be extended to the non-Archimedean case. The main difficulty is the fact that our proof heavily relies on the theory of modules over the ring of differential operators, which does not act  on distributions in the non-Archimedean case. However, in view of  \Cref{thm:AGS} we believe that this difficulty can be overcome. Namely, one can deduce an analog of Theorem \ref{thm:IntroMult}\eqref{it:Bound}  for many  spherical pairs from the following conjecture .
\begin{conj}\label{conj:fin}
Let $G$ be a reductive group defined over a non-Archimedean field $F$ of characteristic $0$ and let $H_1,H_2\subset G$ be its (algebraic) spherical subgroups. Let $\chi_i$ be characters of $H_i\DimaG{(F)}$.
Fix a character $\lambda$ of the Bernstein center $\fz(G\DimaG{(F)})$.

Then the space of distributions which are:

\begin{enumerate}
\item left $(H_1\DimaG{(F)},\chi_1)$-equivariant,
\item right $(H_2\DimaG{(F)},\chi_2)$-equivariant,
\item $(\fz(G\DimaG{(F)}),\lambda)$-eigen,
\end{enumerate}
is finite-dimensional. Moreover, this dimension is uniformly bounded when $\lambda$ varies.
\end{conj}
Note that \Cref{thm:geo} and \Cref{thm:AGS}\eqref{thm:AGS:S} imply that the dimension of (the Zariski closure of) the wave front set of a distribution that satisfies (1-3)  does not exceed $\dim G$.  In many ways the wave front set replaces the singular support, in absence of the theory of differential operators (see, e.g., \cite{Aiz,AD,AGS,AGK}). Thus, in order to prove \Cref{conj:fin}, it is left  to prove analogs of   \Cref{thm:DimSol,thm:FinSol} for the integral system of equations (1-3).

\subsection{Structure of the paper}
In \S \ref{sec:geo}, we prove Theorem \ref{thm:geo} using \Rami{a theorem of Steinberg \cite{Ste76} concerning the Springer resolution.}

In \S \ref{sec:DimSol}, we prove an effective version of Theorem \ref{thm:DimSol}, and then adapt it to algebraic families. We also derive Theorem \ref{thm:FinOrb}.

In \S \ref{sec:Rep}, we derive Theorem \ref{thm:IntroMult}  from Theorem \ref{thm:geo} and \S \ref{sec:DimSol}. We do that by embedding the multiplicity space into a certain space of \DimaD{relative} characters.

In  Appendix \ref{app:dist}, we prove \Cref{lem:PullDist} which computes the pullback of the D-module of distributions with respect to a closed embedding. We use this lemma in \S \ref{sec:DimSol}.

\subsection{Acknowledgements}
We thank Eitan Sayag and Bernhard Kroetz for fruitful discussions.
We thank Joseph Bernstein for telling us the sketch of his proof of Theorem \ref{thm:DimSol}.
\DimaG{We thank Toshiyuki Kobayashi, Alexander Shamov, and the anonymous referee for useful remarks.}

\DimaF{A.A. and D.G. will always be grateful to Joseph Bernstein for introducing them to the amazing world of algebra, for sharing his knowledge, his approaches to problems and his philosophy for already more than half of their lives, and for being a shining example forever.}

\Rami{
\section{Proof of Theorem \ref{thm:geo}}\label{sec:geo}
It is enough to prove the theorem for a reductive group $G$ defined over an algebraically closed field
of characteristic $0$.
Since $S$ includes the zero section of $\DimaB{T^*G \cong }G \times \fg^*$, we have $\dim S \geq \dim G$. Thus, it is enough to prove that $\dim S \leq \dim G$.
Let $\BB$ denote the flag variety of $G$ and $\cN\subset \fg^*$ denote the nilpotent cone.
Since $G$ is reductive, we can identify $$T^*\BB \cong \{ (B,X) \Dima{\in \BB \times \fg^*} \, \vert \, X \in (\mathrm{Lie} B)^{\bot} \}.$$  Recall the Springer resolution $\mu:T^*\BB \to \cN$ defined by $\mu(B,X)=X$ and consider the following diagram.

\begin{equation}\label{eq:smalldiag}
\xymatrixcolsep{5pc}
\xymatrix{
 T^*\BB \times T^*\BB \ar@{->>}[rd]^{\mu \times \mu}  &  & G\times \cN\ar[ld]^{\alpha} \\
  & \cN \times \cN  \ar[d]^{res} & \\
  & \fh_1^* \times \fh_2^* &

}
\end{equation}

Here, $\alpha$ is defined by $\alpha(g,X)=(X,\Ad^*(g^{-1})X)$, and $res$ is the restriction.
Passing to the fiber of $0\in \fh_1^* \times \fh_2^*$, we obtain the following diagram.

\begin{equation}\label{eq:fiberdiag}
\xymatrixcolsep{5pc}
\xymatrix{
 L_1 \times L_2 \ar@{->>}[rd]^{\mu'}  &  & S\ar[ld]^{\alpha'}\\
  & \cN_{\fh_1} \times \cN_{\fh_2}  &
 }
\end{equation}

Here, ${\NC}_{\fh_i}:=\NC \cap \fh_i^{\bot}$ and $L_i:=\{ (B,X) \in T^*\BB \, \vert \, X \in \fh_i^{\bot} \}$
We need to estimate $\dim S$.
We do it using the following lemma.
\begin{lemma}[See \S \ref{sec:lemma} below]\label{lem:tech}
Let $\varphi_i:X_i\to Y$, $i=1,2$, be morphisms of algebraic varieties. Suppose that $\varphi_2$ is surjective. Then there exists $y\in Y$ such that
$$
\dim X_1\leq \dim X_2 + \dim \varphi_1^{-1}(y) - \dim\varphi_2^{-1}(y).
$$
\end{lemma}

By this lemma, applied to $\phi_1=\alp'$ and $\phi_2=\mu'$, it is enough to estimate the dimensions of $L_i$ and of the fibers of $\mu'$ and $\alpha'$.
\begin{lemma}
We have $\dim L_1=\dim L_2 = \dim \BB.$
\end{lemma}
\begin{proof}
Since $H_i$ has finitely many orbits in $\BB$, it is enough  to show that $L_i$ is the union of the conormal bundles to the orbits of $H_i$ in $\BB$.
Let $B\in \BB$, and $\fb=\mathrm{Lie} B$, and identify $T_B\BB \cong \fg/\fb$. Then $T_B(H_i\cdot B)\cong \fh_i /(\fb\cap \fh_i)$ and the conormal space at $B$ to the $H_i$-orbit of $B$ is identified with $ \fb^\bot \cap \fh_i^\bot$.
\end{proof}

Let $(\eta, \Ad^*(g)\eta)\in \mathrm{Im}(\alpha')$. The fiber $(\alpha')^{-1}(\eta, \Ad^*(g)\eta)$ is isomorphic to the stabilizer $G_{\eta}$, and the dimension of the fiber $(\mu')^{-1}(\eta, \Ad^*(g)\eta)$ is twice the dimension of the Springer fiber $\mu^{-1}(\eta)$.
Recall the following  theorem of Steinberg (conjectured by Grothendieck):
\begin{theorem}[{\cite[Theorem 4.6]{Ste76}}]\label{thm:GrConj}
$$
\dim G_\eta-2\dim\mu^{-1}(\eta)=\rk G.
$$
\end{theorem}
Using Lemma \ref{lem:tech}, we obtain for some $(\eta, ad^*(g)\eta)$:
\begin{multline*}
\dim S \leq \dim (L_1\times L_2) + \dim (a')^{-1}(\eta, ad^*(g)\eta) - \dim (\mu')^{-1}(\eta, ad^*(g)\eta) =\\= 2\dim \BB +  \dim G_\eta-2\dim\mu^{-1}(\eta)=2\dim \BB + \rk G = \dim G.
\end{multline*}
\qed

\subsection{Proof of Lemma \ref{lem:tech}}\label{sec:lemma}
Recall that, for a dominant morphism $\varphi: X\to Y$ of irreducible varieties, there exists an open dense $U\subset Y$ such that $\dim X=\dim Y+\dim\varphi^{-1}(y)$ for all $y\in U$ (see, e.~g.,~\cite[Theorem 1.8.3]{RedBook}).
Let $Z$ be an irreducible component of $X_1$ \Dima{of maximal dimension} and $W\subset Y$ be the Zariski closure of $\varphi_1(Z)$. Since $W$ is irreducible, there exists an open dense $U\subset W$ such that
\begin{equation}\label{eq:dim1}
\Dima{\dim X_1 =} \dim Z\leq \dim W+\dim\varphi_1^{-1}(y)
\end{equation}
for all $y\in U$.
Let $V\subset U$ be an open dense subset such that $\varphi_2^{-1}(V)$ intersects those and only those irreducible components $C_1,\ldots,C_j$ of $\varphi_2^{-1}(W)$ that map dominantly to $W$. \Dima{Note that $j>0$ since $\varphi_2$ is surjective.}
Moreover, without loss of generality, we may assume that, for every $1\leq i\leq j$, all fibers over $V$ of the restriction of $\varphi_2$ to $C_i$ are of the same dimension. Since one of these dimensions has to be equal to $\dim\varphi_2^{-1}(V)-\dim V$, we have, that there is an $1\leq i\leq j$ such that, for all $y\in V$,
\begin{equation}\label{eq:dim2}
\dim V=\dim C_i - \dim(\varphi_2|_{C_i})^{-1}(y) \leq \dim\varphi_2^{-1}(V)-\dim\varphi_2^{-1}(y)\leq \dim X_2-\dim\varphi_2^{-1}(y).
\end{equation}
Thanks to $\dim V=\dim W$, taking any $y\in V$, formulas~\eqref{eq:dim1} and~\eqref{eq:dim2} imply the statement. \qed

}
\section{Dimension of the space of solutions of a holonomic system}\label{sec:DimSol}
In this section, we prove an effective version of Theorem \ref{thm:DimSol}, and then adapt it to algebraic families. We also derive Theorem \ref{thm:FinOrb}.

\subsection{Preliminaries}\label{subsec:DimSolPrel}

\subsubsection{D-modules}
In this section, we will use the theory of D-modules \Rami{on algebraic varieties over an arbitrary field $k$ of characteristic zero}.
We will now recall some facts and notions that we will use.
For a good introduction to the algebraic theory of D-modules, we refer the reader to \cite{Ber} and \cite{Bor}. For a short overview, see \cite[Appendix B]{AMOT}.

By a \emph{D-module} on a smooth algebraic variety $X$ we mean a \DimaF{quasi-}coherent sheaf of right modules over the sheaf $D_X$ of algebras of algebraic differential operators. \DimaF{ By a \emph{finitely generated D-module} on a smooth algebraic variety $X$ we mean a coherent sheaf of right modules over the sheaf $D_X$.}
Denote the category of $D_X$-modules by $\cM(D_X)$.

For a smooth affine variety $V$, we denote $D(V):=D_V(V)$.
Note that the category $\cM(D_V)$ of D-modules on $V$ is equivalent to the category of $D(V)$-modules.
We will thus identify these categories.

The algebra $D(V)$ is equipped with a filtration which is called the geometric filtration and defined by the degree of differential operators. The associated graded algebra with respect to this fitration is the algebra $\cO(T^*V)$ of regular functions on the total space of the cotangent bundle of $V$. This allows us to define the \emph{singular support} of a \DimaF{finitely generated} D-module $M$ on $V$ in the following way. \Rami{Choose a good filtration on $M$, i.e. a filtration such that the associated graded module is a finitely-generated module over $\cO(T^*V)$, and define the singular support $SS(M)$ to be the support of this module. One can show that the singular support does not depend on the choice of a good filtration on $M$.}

This definition easily extends to the non-affine case. A \DimaF{finitely generated} D-module $M$ on $X$ is called \emph{smooth} if $SS(M)$ is the zero section of $T^*X$. This is equivalent to being coherent over $\cO_X$ and to being coherent and locally free over $\cO_X$. The Bernstein inequality states that, for any non-zero \DimaF{finitely generated} $M$, we have $\dim SS(M) \geq \dim X$. If the equality holds then $M$ is called \emph{holonomic}.

For a closed embedding $i:X \to Y$ of smooth affine algebraic varieties, define the functor $i^!:\cM(D_Y)\to \cM(D_X)$ by $i^!(M):=\{m\in M \, \vert \, I_Xm=0\},$
where $I_X\subset \cO(Y)$ is the ideal of all functions that vanish on $X$.
\DimaC{It has a left adjoint functor $i_*:\cM(D_X) \to \cM(D_Y)$, given by tensor product with $i^!(D_Y)$. The functor $i_*$ is an equivalence of categories between $\cM(D_X)$ and the category of $D_Y$-modules supported in $X$. Both $i_*$ and $i^!$ map holonomic modules to holonomic ones.}

If $V$ is an affine space then the algebra $D(V)$ has an additional filtration, called the Bernstein filtration. It is defined by $\deg(\partial/\partial x_i)=\deg(x_i)=1,$  where $x_i$ are the coordinates in $V$. This gives rise to the notion of Bernstein's singular support, that we will denote $SS_b(M)\subset T^*V\cong V\oplus V^*$. It is known that $\dim SS(M)=\dim SS_b(M)$.

We will also use the theory of analytic D-modules. By an \emph{analytic D-module} on a smooth \Rami{complex} analytic manifold $X$ we mean a coherent sheaf of right modules over the sheaf $D_X^{An}$ of algebras of  differential operators with analytic coefficients.
All of the above notions and statements, except those concerning the Bernstein filtration, have analytic  counterparts. In addition, all smooth analytic D-modules of the same rank are isomorphic.

\subsubsection{Distributions}
We will use the theory of distributions on differentiable manifolds and the theory of tempered distributions on real algebraic manifolds, see e.g. \cite{Hor,AGSc}.
For \DimaG{an  algebraic manifold $X$ defined over $\R$, we denote the space of \DimaG{real valued distributions on $X(\R)$ by $\dist(X(\R),\R):=(C_c^{\infty}(X(\R),\R))^*$} and the space of real valued tempered  distributions (a.k.a. Schwartz distributions) by $\Sc^*(X\DimaG{(\R)},\R):=(\Sc(X\DimaG{(\R),\R}))^*$. Similarly, denote by $\dist(X(\R)):=\dist(X(\R),\C)$ the space of complex valued distributions on $X(\R)$ and by $\Sc^*(X(\R)):=\Sc^*(X(\R),\C)$ the space of complex valued tempered distributions on $X(\R)$. Also,
for an algebraic bundle $\cE$ (complex or real) over $X$ we denote $\dist(X\DimaG{(\R)},\cE):=(C_c^{\infty}(X\DimaG{(\R)},\cE))^* $ and  $\Sc^*(X\DimaG{(\R)},\cE):=(\Sc(X\DimaG{(\R)},\cE))^*$.}

\DimaG{
The algebra $D(X)$  acts on the spaces $\dist(X\DimaG{(\R),\R})$ and $\Sc^*(X\DimaG{(\R),\R})$.
Thus, for affine $X$ we can consider these spaces as $D_{X}$-modules and $\dist(X\DimaG{(\R),\R})$  also as an analytic $D_X$-module.
Similarly, we will regard the spaces $\dist(X(\R))$ and $\Sc^*(X(\R))$ as $D_{X_{\C}}$-modules, where $X_{\C}$ denotes the complexification of $X$.
For non-affine $X$, define the $D_X$-module $\Sc^*_{X,\R}$ by $\Sc^*_{X,\R}(U):=\Sc^*(U(\R),\R)$.
It is easy to see that this sheaf is quasi-coherent. Denote by $\Sc^*_{X}$ the
$D_{X_{\C}}$-module obtained by
complexification of $\Sc^*_{X,\R}$. }

We define the singular support of a \DimaG{tempered} distribution to be the singular support of the D-module it generates. \Rami{It is well known  that this definition \DimaB{is equivalent to} Definition \ref{def:SISDist}.}
We say that a distribution is holonomic if it generates a holonomic D-module.

\begin{lem}[See \Cref{app:dist}]\label{lem:PullDist}
Let $i:X \to Y$ be a closed embedding of smooth affine \DimaG{ algebraic varieties defined over $\R$}.
Then
$$\dist(X\DimaG(\R))\cong i^!(\dist(Y\DimaG(\R))) \text{ and }\Sc^*(X\DimaG(\R))\cong i^!(\Sc^*(Y\DimaG(\R))).$$
\end{lem}

\begin{lem}\label{lem:Free}
Let $M$ be a smooth $\DimaG{D(\bA^n_{\C} )}$-module of rank $r$. \Rami{Embed the space $An(\C^n)$ of analytic functions on $\C^n$ into $\dist(\R^n)$ using the Lebesgue measure.} Then $$\Hom_{\DimaG{D(\bA^n_{\C} )}}(M,\dist(\R^n))=\Hom_{\DimaG{D(\bA^n_{\C} )}}(M,An(\C^n)) \text{ and }\dim \Hom_{\DimaG{D(\bA^n_{\C} )}}(M,\dist(\R^n))= r.$$
\end{lem}
\begin{proof}
Let $M_{An}:=M \otimes_{\cO(\C^n)}An(\C^n)$ and $\DimaG{D_{An}(\bA^n_{\C} )}:=\DimaG{D(\bA^n_{\C} )} \otimes_{\cO(\C^n)}An(\C^n)$   be the analytizations of $M$ and $\DimaG{D(\bA^n_{\C} )}$. Then $$\Hom_{\DimaG{D(\bA^n_{\C} )}}(M,\dist(\R^n)) \cong \Hom_{\DimaG{D_{An}(\bA^n_{\C} )}}(M_{An},\dist(\R^n)).$$
Since $M_{An}$ is also smooth,
$M_{An} \cong An(\C^n)^r$. Thus it is left to prove that
$$\Hom_{\DimaG{D_{An}(\bA^n_{\C} )}}( An(\C^n),\dist(\R^n))=\Hom_{\DimaG{D_{An}(\bA^n_{\C} )}}( An(\C^n),An(\C^n))$$ and the latter space is one-dimensional.  This follows from the fact that a distribution with vanishing partial derivatives is a multiple of the Lebesgue measure.
\end{proof}

\begin{cor}\label{cor:SmoothDist}
If a distribution generates a smooth $D$-module, then it is analytic.
\end{cor}

\Rami{
\subsubsection{Lie algebra actions}

\begin{defn}Let $X$ be an algebraic manifold \DimaG{defined} over a field $k$ and $\fg$ be a  Lie algebra over $k$.
\begin{enumerate}[(i)]
\item An action of $\fg$ on $X$ is a Lie algebra map from $\fg$ to the algebra of algebraic vector fields on $X$.
\item Assume that $X$ is affine, fix an action of $\fg$ on $X$ and let $\cE$  be an algebraic vector bundle on $X$. Let $M$ be the space of global \DimaB{regular} {(algebraic)} sections of $\cE$. An action of $\fg$ on $\cE$ is a linear map $T:\fg \to \End_{k}(M)$ such that, for any $\alp\in \fg , \, f\in \cO(X), \, v \in M$, we have
    $$T(\alp)(fv)=(\alp f) v+fT(\alp)v.$$
\item The definition above extends to non-affine $X$ in a straightforward way.
\end{enumerate}
\end{defn}
}
\subsubsection{Weil representation}

\begin{defn}
Let $V$ be a finite-dimensional real vector space. Let $\omega$ be  the standard symplectic form on $V \oplus V^*$. Denote by $p_V:V \oplus V^* \to V$ and $p_{V^*}:V \oplus V^*\to V^*$ the natural projections.
Define an action of the symplectic group $\DimaG{\Sp(V \oplus V^*)}$ on the algebra $D(V)$
by
$$(\partial_v)^g:=\pi(g)(\partial_v):=p_{V^*}(g(v,0)) + \partial_{p_{V}(g(v,0))}, \quad w^g:=\pi(g)w:=p_{V^*}(g(0,w)) + \partial_{p_{V}(g(0,w))}$$
where $v\in V, \, w\in V^*, \, \partial_v$ denotes the derivative in the direction of $v$, and elements of $V^*$ are viewed as linear polynomials and thus differential operators of order zero.
For a  $D(V)$-module $M$ and an element $g\in \Sp(V \oplus V^*)$, we will denote by $M^g$ the $D(V)$-module obtained by twisting the action of $D(V)$ by $\pi(g)$.
\end{defn}

Since the above action of $\Sp(V \oplus V^*)$ preserves the Bernstein filtration on $D(V)$,  the following lemma holds.
\begin{lem}\label{lem:SympSupp}
For a  \DimaF{finitely generated} $D(V)$-module $M$ and $g\in \Sp(V \oplus V^*)$ we have $SS_{b}(M^g)=gSS_{b}(M)$.
\end{lem}

\begin{thm}[\cite{Weil}]
There exists a two-folded cover $p:\widetilde{\Sp}(V \oplus V^*)\to \Sp(V \oplus V^*)\DimaG(\R)$ and a representation $\Pi$ of $\widetilde{\Sp}(V \oplus V^*)$ on the space $\Sc^*(V)$ of tempered distributions on $V$ such that, for any $\alpha \in D(V)$, $g\in \widetilde{\Sp}(V \oplus V^*)$, {$\xi\in\Sc^*(V)$,} we have$$\Pi(g)(\xi\alpha)=(\Pi(g)\xi)\alpha^{p(g)}.$$
\end{thm}

\begin{cor}\label{cor:Weil}
We have an isomorphism of  $D(V)$-modules $\Sc^*(V)^g \cong \Sc^*(V)$ for any $g\in \Sp(V \oplus V^*)\DimaG(\R)$.
\end{cor}
In fact, this corollary can be derived directly from the Stone-von-Neumann theorem.
\subsubsection{Flat morphisms}

\begin{lemma}\label{lem:flat}
Let $\phi:X \to Y$ be a proper morphism of algebraic varieties \DimaG{defined} \Rami{over a field $k$} and $\cM$ be a coherent sheaf on $X$. Then there exists an open dense $U\subset Y$ such that $\cM|_{\phi^{-1}(U)}$ is flat over $U$.
\end{lemma}
\begin{proof}
By \cite[Th{\'e}or{\`e}me II.3.I]{EGA4.3}, the set $V$ of {scheme-theoretic} points $x\in X$ for which $\cM$ is $\phi$-flat at $x$ is open in $X$. Since $\phi$ is proper, the set $Z:=\phi(X\setminus{V})$ is closed in $Y$. Note that $\cM$ is flat over $U:=X\setminus{Z}$, since $\phi^{-1}(U)\subset V$. Moreover, $U$ contains the generic points of the irreducible components of $Y$. Hence, $U\subset Y$ is dense.
\end{proof}

\begin{lemma}[{See, e.~g., \cite[Corollary on p.~50]{Mumford}}]\label{lem:loc_const}
Let $\phi:X \to Y$ be a proper morphism of algebraic varieties \DimaG{defined over a field $k$}  and $\cM$ be a coherent sheaf on $X$ that is flat over $Y$. \Rami{For a point $y\in Y$, let $\cM_y$ denote the pullback of $\cM$ to $\phi^{-1}(y)$.}
Then the function
$$y \mapsto \chi(\cM_y)=\sum_{i=0}^{\infty}(-1)^i\dim_{k(y)} \oH^i(\cM_y)$$
is locally constant.
\end{lemma}
\Rami{
\begin{cor}\label{cor:loc_const}
Let $Y$ be an algebraic variety \DimaG{defined over a field $k$} and $\cM$ be a coherent sheaf on $Y \times \bP^n$. Then there exists an open dense $U\subset Y$ such that the Hilbert polynomial\footnote{For the definition of Hilbert polynomial see \cite[Chapter III, Exercise 5.2]{Har}.} of $\cM_y$ does not depend on $y$ as long as $y\in U$.
\end{cor}
}
\subsection{Dimension of the space of solutions of a holonomic system}\label{subsec:DimSol}

\begin{defn} \DimaG{Let $k$ be a field of characteristic zero.}
\begin{enumerate}[(i)]
\item
Let $M$ be a \DimaF{finitely generated} $D$-module over an affine space \DimaG{$\mathbb{A}_k^n$ over $k$}. Let $F^i$ be a good filtration on $M$ with respect to the Bernstein filtration on the ring $D_{\DimaG{\mathbb{A}_k^n}}$. Let $p$ be the corresponding Hilbert polynomial of $M$, i.e. $p(i)=\dim F^i$ for large enough $i$. Let $\DimaG{d}$ be the degree of $p$ and $a_{\DimaG{d}}$ be the leading coefficient of $p$. Define the Bernstein degree of $M$ to be $\deg_b(M):=\DimaG{d}!a_{\DimaG{d}}$. It is well-known that $\DimaG{d}$ and $a_{\DimaG{d}}$ do not depend on the choice of good filtration $F^i$.
\item Let $M$ be a \DimaF{finitely generated} $D$-module over a smooth algebraic variety $X$ \DimaG{defined over $k$}. Let $X=\bigcup_{i=1}^l U_i$ be an open affine cover of $X$ and let $\phi_i:U_i \hookrightarrow \DimaG{\mathbb{A}_k^{n_i}}$ be closed embeddings. Denote
$$\deg_{\{(U_i,\phi_i)\}}(M):=\sum_{i=1}^l \deg_b((\phi_i)_*(M\DimaC{|_{U_{i}}})).$$
Define the global degree of $M$ by $\deg(M):=\min  \deg_{\{(U_i,\phi_i)\}}(M)$, where the minimum is taken over the set of all possible affine covers and embeddings.
\end{enumerate}
\end{defn}

In this \Rami{subsection}, we prove
\begin{thm}\label{thm:FinSol}
Let $X$ be \DimaG{an algebraic manifold defined over $\R$}. Let $M$ be a holonomic right $D_{\DimaG{X_{\C}}}$-module. Then $\dim \Hom(M,\Sc^*(X)) \leq \deg(M)$.
\end{thm}

We will need the following geometric lemmas

\begin{lemma}\label{lem:PrFin}
Let $V$ be a vector space \DimaG{over an algebraically closed field}, $L\subset V$ be a subspace and $C\subset V$ be a closed conic algebraic subvariety such that $L\cap C = \{0\}$. Then the projection $p:C \to V/L$ is a finite map.
\end{lemma}
\begin{proof}
By induction, it is enough to prove the case $\dim L =1$. Choose coordinates $x_1,\dots,x_n$ on $V$ such that the coordinates $x_1,\dots,x_{n-1}$ vanish  on $L$. Let $p$ be a homogeneous polynomial that vanishes on $C$ but not on $L$. Write $p=\sum_{i=1}^{\DimaG{d}} g_ix_n^i$, where each $g_i$ is a homogeneous polynomial of degree $\DimaG{d}-i$ in $x_1,\dots,x_{n-1}$. Then $x_n|_C$ satisfies a monic polynomial equation with coefficients in $\cO(V/L)$.
\end{proof}

\begin{lemma}\label{lem:Lag}
Let $W$ be a $2n$-dimensional \DimaG{real} symplectic vector space, and $C\subset \DimaH{W_{\C}}$ be a closed conic   subvariety of dimension $n$. Then there exists a \DimaG{real} Lagrangian subspace $L\subset W$ such that $\DimaG{L_{\C}\cap C} = \{0\}$.
\end{lemma}
\begin{proof}
Let $\cL$ denote the variety of all Lagrangian subspaces of $W$. Note that $\dim \cL = n(n+1)/2$. Let $P(C)\subset \bP(W_{\DimaH{\C}})$ be the projectivizations of $C$ and $W$.
Consider the configuration space $$X:=\{(x,L)\in P(C) \times \cL \, \vert \, x \subset L\}.$$
\DimaG{Since $\cL$ is smooth, irreducible, and has a real point, it is enough}
 to show that $\DimaH{\overline{p(X)}} \neq \cL$ where $p:X \to \cL$ is the projection. Let $q:X\to P(C)$ be the other projection. Note that $\dim q^{-1}(x)=n(n-1)/2$ for any $x \in P(C)$. Thus $$\dim X = n(n-1)/2+n-1<n(n+1)/2=\dim \cL,$$
and thus $p:X \to \cL_{\DimaH{\C}}$ cannot be \DimaH{dominant}.
\end{proof}

\begin{cor}\label{cor:PrLag}
Let $V$ be a \DimaG{real} vector space of dimension $n$. Consider the standard symplectic form on $V\oplus V^*$. Let $C\subset V_{\DimaH{\C}}\oplus V_{\DimaH{\C}}^*$ be a closed conic subvariety of dimension $n$\DimaG{, defined over $\R$}. Let $p:V_{\DimaH{\C}}\oplus V_{\DimaH{\C}}^*\to V_{\DimaH{\C}}$ denote the projection. Then there exists a linear symplectic automorphism $g \in \Sp(V\oplus V^*)\DimaG{(\R)}$ such that $p|_{gC}$ is a finite map.
\end{cor}
\begin{proof}
By \Cref{lem:Lag} there exists a Lagrangian subspace $L\subset V \oplus V^*$ such that $\DimaG{L_{\C}\cap C} = \{0\}$.
 Since the action of $\Sp(V\oplus V^*)\DimaG{(\R)}$ on Lagrangian subspaces is transitive, there exists $g \in \Sp(V\oplus V^*)\DimaG{(\R)}$ such that \DimaC{$V^*=gL$  and thus $\DimaG{gC\cap V_{\C}^*}=\{0\}$.}
From \Cref{lem:PrFin} we get that $p|_{gC}$ is a finite map.
\end{proof}

\begin{proof}[Proof of \Cref{thm:FinSol}]
Let $X=\bigcup_{i=1}^l U_i$ be an open affine cover of $X$ and let $\phi_i:U_i \hookrightarrow \mathbb{A}_{\DimaH{\C}}^{n_i}$ be closed embeddings. Clearly $$\dim \Hom (M,\Sc^*_{\DimaG{X}})\leq \sum_{i=1}^l \dim \Hom (M|_{\DimaG{(U_i)_{\C}}},\Sc^*(U_i\DimaG{(\R))}).$$ By  \Cref{lem:PullDist} $$\Hom (M|_{\DimaG{(U_i)_{\C}}},\Sc^*(U_i\DimaG{(\R)})) \cong \Hom (M|_{\DimaG{(U_i)_{\C}}},\phi_i^!(\Sc^*(\R^{n_i}))\cong\Hom ((\phi_i)_*(M|_{\DimaG{(U_i)_{\C}}}),\Sc^*(\R^{n_i})).$$

Thus it is enough to show that for any holonomic $D$-module $N$ on an affine space $\bA^n_{\DimaH{\C}}$ we have
$$\dim \Hom (N,\Sc^*(\R^{n})) \leq \deg_b(N).$$

Let $C\subset \bA^{2n}_{\DimaH{\C}}$ be the singular support of $N$ with respect to the Bernstein filtration.
By \Cref{cor:PrLag}, there exists $g \in \Sp_{2n}\DimaG{(\R)}$ such that $p|_{gC}$ is a finite map, where $p:\bA_{\DimaH{\C}}^{2n}\to \bA_{\DimaH{\C}}^n$ is the projection on the first $n$ coordinates. By Corollary \ref{cor:Weil}  we have $$\dim \Hom(N,\Sc^*(\R^n)) = \dim \Hom(N^g,\Sc^*(\R^n)^g)=\dim \Hom(N^g,\Sc^*(\R^n)).$$
By Lemma \ref{lem:SympSupp} we have $SS_b(N^g)=gC$. Let $F$ be a good filtration on $N^g$ (with respect to the Bernstein filtration on $D(\bA_{\DimaH{\C}}^n)$). We see that $\Gr N^g$ is finitely generated over $\cO(\bA_{\DimaH{\C}}^n)$, and thus so is $N^g$. Thus $N^g$ is a smooth $D$-module. Note that $\rk_{\cO(\bA^n)} N^g \leq \deg_b N^g=\deg_b N$. By Lemma \ref{lem:Free}  $\dim \Hom(N^g,\Sc^*(\R^n)) \leq \rk_{\cO(\bA_{\DimaH{\C}}^n)} N^g$.
\end{proof}

\subsection{Families of $D$-modules}\label{subsec:DimSolRel}
\Rami{In this section we discuss families of $D$-modules on algebraic varieties over an arbitrary field $k$ of zero characteristic.}
\begin{notn}
Let $\phi:X \to Y$ be a map of algebraic varieties and $\cM$ be a quasi-coherent sheaf of $\cO_X$-modules. For any $y\in Y$, denote by $\cM_y$ the pullback of $\cM$ to $\phi^{-1}(y)$.
\end{notn}

\begin{defn}Let $X,Y$ be smooth algebraic varieties.
\begin{itemize}
\item If $X$ and $Y$ are affine we define the algebra $D(X,Y)$ to be $D(X)\otimes_k \cO(Y)$.
\item Extending this definition we obtain a  sheaf of algebras $D_{X,Y}$ on $X \times Y$.
\item \Rami{By a family of $D_X$-modules parameterized by $Y$, we mean a sheaf of right }modules over the sheaf of algebras $D_{X,Y}$ on $X \times Y$ which is quasicoherent as a sheaf of $\cO_{X\times Y}$-modules.

\item We call a family of $D_X$-modules parameterized by $Y$ coherent if it is locally finitely generated as a $D_{X,Y}$-module.

\item For a family $\cM$ of $D_X$-modules parameterized by $Y$ and a point $y\in Y$, we  call $\cM_y$ the specialization of $\cM$ at $y$ and consider it with the natural structure of a $D_X$-module.
\item We say that a coherent family $\cM$ is holonomic if every specialization is holonomic.
\end{itemize}

\end{defn}

\begin{thm}\label{thm:DegBound}
Let $X,Y$ be smooth algebraic varieties and $\cM$  be a family of $D_X$-modules parametrized by $Y$. Then $\deg\cM_y$ is bounded when $y$ ranges over the $k$-points of $Y$.
\end{thm}

\begin{proof

Without loss of generality, we can assume that $X=\bA^n$ and $Y$ is an affine variety, and prove that $\deg_b(\cM_y)$ is bounded. We will prove this by induction on $\dim Y$.

The Bernstein filtration on $D(\bA^n)$ gives rise to a filtration on $D(\bA^n,Y)$. Choose a filtration $F$ on $\cM$ which is good with respect to this filtration and let $N:=\Gr{\cM}$, considered as a graded $\cO(\bA^{2n}\times Y)$-module.
Associate to $N$ a coherent sheaf $\cN$ on $\bP^{2n-1}\times Y$.
Let $\cN_y$ be the pullback  of $\cN$ under the embedding of $\bP^{2n-1}$ into $\bP^{2n-1}\times Y$ given by $x \mapsto (x,y)$.
By definition, the Hilbert polynomial of $\cM_y$ with respect to the filtration induced by $F$ is the Hilbert polynomial of $\cN_y$.
\Rami{By Corollary \ref{cor:loc_const}, there exists an open dense subset $U \subset Y$ such that the Hilbert polynomial of $\cN_y$ does not depend on $y$ as long as $y\in U$.}
By the induction hypothesis, $\deg_b(\cM_y)$ is bounded on $Y \setminus U$, and thus bounded on $Y$.
\end{proof}

For an application of this theorem we will need the following lemma.
\DimaD{
\begin{lem}\label{lem:action}
Let a real Lie algebra $\fg$ act on \DimaG{an algebraic manifold $X$ defined over $\R$} and on \DimaG{a complex} algebraic vector bundle $\cE$ on $X$. Fix a natural number $n$ and let $Y$ be the variety of all representations of $\fg$ on $\C^n$. Then there exists a coherent family $\cM$ of $D_X$-modules parameterized by $Y$ such that, for any $\tau \in Y$, we have
\begin{enumerate}
\item $\Hom_{\fg}(\tau,\Sc^*(X(\R),\cE))=\Hom_{D_X}(\cM_{\tau},\Sc^*_X).$
\item  The singular support of $\cM_{\tau}$ (with respect to the geometric filtration) is included in $$\{(x,\phi) \in T^*X \, \vert \,  \forall \alpha \in \fg \text{ we have } \langle \phi, \alpha(x)\rangle =0 \}. $$
\end{enumerate}

\end{lem}
\begin{proof}
It is enough to prove the lemma for affine $X$. Let $N$ be the coherent sheaf of the regular {(algebraic)} sections of $\cE$ (considered as a sheaf of  $\cO_{\DimaG{X_{\C}}}$-modules). Let $\cN$ be the pullback of $N$ to $\DimaG{X_{\C}\times Y}$. Let $\cN':=\cN\otimes_{\cO_{\DimaG{X_{\C}\times Y}}}D_{\DimaG{X_{\C}, Y}}\otimes_{\C}\C^n$, and  $\cN''\subset \cN'$ be the \Rami{$D_{\DimaG{X_{\C}, Y}}$-submodule generated} by elements of the form $$\alpha n \otimes 1 \otimes v+ n \otimes \xi_{\alpha} \otimes v + n\otimes  f_{\alpha}(v),$$
where $\alpha\in \fg$, $\xi_{\alpha}$  is the vector field on $X$ corresponding to $\alpha$,  and $f_{\alpha}(v)\in D_{\DimaG{X_{\C}, Y_{\C}}}\otimes_{\C}\C^n$ is the $\C^n$-valued regular function on $\DimaG{X_{\C}\times Y}$ given by $f_{\alpha}(v)(x,\tau)=\tau(\alpha)v$. Then $\cM:=\cN'/\cN''$ satisfies the requirements.\end{proof}
}
\Cref{thm:FinSol,thm:DegBound,lem:action} imply \Cref{thm:FinOrb}.

\section{Proof of \Cref{thm:main,thm:IntroMult}}\label{sec:Rep}

In this section, we derive  \Cref{thm:main,thm:IntroMult}  from Theorem \ref{thm:geo} and \S \ref{sec:DimSol}. We do that by embedding the multiplicity space into a certain space of \DimaD{relative} characters.

\subsection{Preliminaries}

For  \DimaG{a reductive group $G$ defined over $\R$}, we denote by $Irr(G\DimaG{(\R)})$ the collection of irreducible admissible smooth \Fre representation of $G\DimaG{(\R)}$ of moderate growth. We refer to \cite{CasGlob,Wal} for the background on these representations.

\begin{theorem}[See {\cite[Theorem 4.2.1]{Wal}}]\label{thm:CentFin}
\Rami{The center $\fz(\cU(\fg))$ of the universal enveloping algebra of the \DimaG{complexified} Lie algebra of $G$ acts finitely on every admissible smooth \Fre representation $\pi$ of $G$ of moderate growth.
This means that there exists an ideal in $ \fz(\cU(\fg))$ of finite codimension that annihilates $\pi$.
}
\end{theorem}

\begin{lem}[{\cite[Theorem 1.2 and Corollary 1.4]{Adams}}]\label{lem:GK}
For any \DimaG{reductive group $G$ defined over $\R$}, there exists an involution $\theta$ of $G$ such that, for any $\pi\in \Irr(G\DimaG{(\R)})$, we have $\hat \pi \cong \pi^{\theta}$.
\end{lem}

\begin{thm}[Casselman embedding theorem, see {\cite[Proposition 8.23]{CM}}]\label{thm:CasSubRep}
Let $G$ be \DimaG{a reductive group $G$ defined over $\R$} and  $P$ be a minimal parabolic subgroup of $G$. Let $\pi\in Irr(G\DimaG{(\R)})$.   Then there exists a
finite-dimensional
representation $\sigma$ of $P$ and an epimorphism $Ind_{P\DimaG{(\R)}}^{G\DimaG{(\R)}}(\sigma) \twoheadrightarrow \pi$.
\end{thm}

\subsection{Proof of Theorem \ref{thm:main} and Proposition \ref{prop:S}}\label{subsec:PfMain}

Theorem \ref{thm:main} follows from \Cref{thm:geo} and Proposition \ref{prop:S}.
{\sloppy
\begin{proof}[Proof of Proposition \ref{prop:S}]
Let $\xi$ be a \DimaD{relative} character of a smooth admissible \Fre  representation $\pi$ of $G\DimaG{(\R)}$ of moderate growth with respect to a pair of subgroups $(H_1\DimaG{(\R)},H_2\DimaG{(\R)})$ and their characters $\chi_1,\chi_2$. By \Cref{thm:CentFin}, \Rami{  there exists an ideal $I \subset \fz(\cU(\fg))$ of finite codimension that annihilates $\pi$ and thus  annihilates $\xi$.} For any element $z \in \fz(\cU(\fg))$, there exists a polynomial $p$ such that $p(z)\in I$ and thus $p(z)\xi=0$. This implies that the symbol of any  $z \in \fz(\cU(\fg))$ of positive degree vanishes on the singular support of $\xi$. It is well-known  that
the joint  zero-set of these symbols over each point $g\in G$ is the nilpotent cone $\cN(\fg^*)$. Since $\xi$ is $(\DimaC{\fh}_1\times \DimaC{\fh}_2, \chi_1\times \chi_2)$-equivariant,
this implies that the singular support of $\xi$ lies in $S$.
\end{proof}}
\subsection{Proof of Theorem \ref{thm:IntroMult}}\label{subsec:PfIntroMult}
Part \eqref{it:Fin} follows immediately from \Cref{thm:FinOrb} and the Casselman embedding theorem. If $G$ is quasi-split then so does part \eqref{it:Bound}.
For the proof of part \eqref{it:Bound} in the general case, we will need the following lemma.

\begin{lem}\label{lem:SphCharBound}
\Rami{Let $G$ be \DimaG{a reductive group $G$ defined over $\R$}  and $H_1,H_2$ be spherical subgroups. Let $Y=\mathrm{Spec}(\fz(\cU(\fg)))\Andrey{\times Y_1\times Y_2}$, \Andrey{where $Y_i$ is the variety of characters of $\ha_i=\mathrm{Lie\,}H_i$}. For any $\lambda\in Y(\C)$, define $U_{\lambda,\Andrey{\chi_1,\chi_2}}:=\Sc^*(G\DimaG{(\R)})^{\Andrey{\ha_1\times \ha_2,(\chi_1,\chi_2)},(\fz(\cU(\fg)),\lambda)}$ to be the space of tempered distributions on $G$ that are left \Andrey{$\chi_1$-equivariant with respect to $\ha_1$, right $\chi_2$-equivariant with respect to $\ha_2$} and are eigendistributions with respect to the action of $\fz(\cU(\fg))$ with eigencharacter $\lambda$. Then $\dim  U_{\lambda,\chi_1,\chi_2}$ is bounded over $Y(\C)$}.
\end{lem}

\begin{proof}
\Rami{
Let us construct a family of $D(G)$-modules $\cM$ parameterized by $Y$. For any $\alpha \in \fg$, let $r_\alpha$ and $l_\alpha$ be the corresponding right and left invariant vector fields on $G$ considered as  elements in  $D(G,Y)$. For any $\beta \in \fz(\cU(\fg))$, \Andrey{$\alpha_i\in\ha_i$, let $f_\beta,g^i_{\alpha_i}$ be the functions on $Y$ that send $(\mu,\gamma_1,\gamma_2)\in Y$ to $\mu(\beta), \gamma_i(\alpha_i)$ respectively}. Let also $d_\beta$ be the differential operator on $G$ corresponding to $\beta$, \DimaC{such that
$d_{\beta}\xi=\beta\xi$ for any distribution $\xi$ on $G\DimaG{(\R)}$}.
We consider \Andrey{$d_\beta,r_{\alpha_1},l_{\alpha_2},f_\beta,g^{i}_{\alpha_i}$} as elements of $D(G,Y)$. Let $I \subset D(G,Y)$ be the ideal generated by \DimaC{$r_{\alpha_1}\Andrey{-g^{1}_{\alpha_1}}$, $l_{\alpha_2}\Andrey{-g^{2}_{\alpha_2}}$ and $f_\beta \DimaC{-} d_\beta$ where $\alpha_i\in\fh_i$} and $\beta\in \fz(\cU(\fg))$. Define $\cM:=D(G,Y)/I$.

It is easy to see that $U_{\lambda,\chi_1,\chi_2} \cong \Hom(\cM_{(\lambda,\chi_1,\chi_2)},\Sc^*(G\DimaG{(\R)}))$. As in the proof of Proposition \ref{prop:S}, the singular support of $\cM_{(\lambda,\chi_1,\chi_2)}$ lies in $S$, for any $\lambda,\chi_1,\chi_2$. By Theorem \ref{thm:geo}, $\cM_{(\lambda,\chi_1,\chi_2)}$ is holonomic and, therefore, $\cM$ is holonomic. By Theorem \ref{thm:FinSol}, we have $\dim U_{\lambda,\chi_1,\chi_2} \leq \deg \cM_{(\lambda,\chi_1,\chi_2)}$. By \Cref{thm:DegBound}, $\deg \cM_{(\lambda,\chi_1,\chi_2)}$ are bounded.}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:IntroMult}\eqref{it:Bound}]

We choose an involution $\theta$ as in \Cref{lem:GK}, let $H_1:=H, \, H_2:=\theta(H)$, and  define the spaces $U_{\lambda}$ as in \Cref{lem:SphCharBound}.

Now let $\pi\in \Irr(G\DimaG{(\R)})$ \Andrey{and let $\chi$ be a character of $\fh$} such that $(\pi^*)^{\fh,\chi}\neq 0$. Let ${\lambda}$ stand for the infinitesimal character of $\pi$. By \Cref{lem:GK}, $(\hat \pi^*)^{d\theta(\fh),\Andrey{d\theta(\chi)}}\neq 0$. Fix a non-zero $\phi \in (\hat\pi^*)^{{d\theta(\fh),\Andrey{d\theta(\chi)}}}$. Then $\phi$ defines an embedding $(\pi^*)^{\fh,\Andrey{\chi}} \hookrightarrow U_{\lambda,\Andrey{\chi,d\theta(\chi)}}$ by $\psi \mapsto \xi_{\psi,{\phi}}$, where $\xi_{\psi,{\phi}}$ is the \DimaD{relative} character, which is defined by $\xi_{\psi,{\phi}}(f):=\langle \psi , \pi(f) \phi\rangle$. Thus, $\dim (\pi^*)^{\fh,\Andrey{\chi}} \leq \dim U_{\lambda,\Andrey{\chi,d\theta(\chi)}}$, which is bounded by \Cref{lem:SphCharBound}.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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