siegel parabolic subgroup

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Coadjoint Orbits of Siegel Parabolic Subgroups Hongyu HE Department of Mathematics, Louisiana State University Baton Rouge, LA 70803 email: hongyu@math.lsu.edu Let P +(n) be the Siegel parabolic subgroup of O(n,n), and P (n) be the Siegel parabolic subgroup of Sp 2n(R). The intersection of any two parabolic subgroups contains a subgroup of $G$ that is conjugate to $T=B\cap N$. This ag is called a complete ag if r = n and dim(Wi) = i for each i.The Borel subgroups of GL(n;F) are the stabilizers of complete ags. De ne I1 ; = (: G! a minimal parabolic subgroup. In particular, in this case (1 2,)=1bythe functional equation. Resume of the main results. The notion of "minimal parabolic" applies, meaningfully, more generally than does the notion of "Borel subgroup/subalgebra". (i) If nis even then HomH( m,C) = HomP( ,C) C (ii) If nis odd then HomH( m,C) = HomU(n)( ,C) = 0. symplectic group, while the other two are along the Siegel parabolic subgroup or Jacobi parabolic subgroup. Euler . Alert. induced from viewed as a representation on the Siegel parabolic subgroup. 96 (1972), 296--317. Let G denote the special unitary group SU (2, 2) and G has two realization G1, G2 for example. In this paper we prove the following result. In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form). Let V > t .F. Let (;V ) be an admissible, smooth, rcFhet representation of M, and let 2a0 C ( a = Lie(A)). Let be an irreducible, admissible representation of , be the usual Jacquet module with respect to the Siegel parabolic subgroup. We recall some information from [9]. Save. We call P the Siegel maximal parabolic subgroup of Gn. (, ). Stack Overflow for Teams is moving to its own domain! Similarly, we define the twisted Jacquet module of associated to as follows In fact, admits an action of H. C. L. Siegel, Einfuhrung in die Theorie der Modulfunktionen n . The element (x 0 1) is in Sp 2n if and only if x= xt. In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form). Induced representations 2. classical Fourier . Preprint, 2019. We show that e a (B). /Length 2520 (ii)A Borel subgroup is parabolic. where (s, g) is an Eisenstein series induced from a character of the Siegel parabolic subgroup, is a cuspidal automorphic form, is an n-by-n symmetric matrix determining an dimensional orthogonal space Vt, and O'^(g) is the theta kernel Received by the editors February 28, 2012 and, in revised form, April 6, 2012 and May 2, 2012. By [20], P is self-associate, unless Gn = SO2n and n is odd. According to our current on-line database, Yasuro Gon has 3 students and 3 descendants . Let P be the maximal standard parabolic k-subgroup of Gn corresponding to the subset nfng of the set of simple roots . References . We solve the recursive formula by the method of Zagier . The standard Borel subgroup is the stabilizer of the . To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Internal modules of parabolic subgroups. In this paper we show that the Eisenstein series with respect to the Siegel parabolic subgroup corresponds to the Eisenstein series with respect to the Jacobi parabolic subgroup by theta correspondences. arXiv:1509.00900v2 [math.NT] 28 Jan 2016 A LOCAL CONVERSE THEOREM FOR U(2,2) QING ZHANG Abstract. Dissertation: Generalized Whittaker functions on SU (2,2) with respect to the Siegel parabolic subgroup Mathematics Subject Classification: 11Number theory Advisor 1: Takayuki Oda Students: Click here to see the students listed in chronological order. We identify M with via the map (8) carries an action of M, and thus an action of via this isomorphism. In mathematics , the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product . Levi subgroup of Siegel parabolic of GSpin. The constant term (along the "Siegel radical") of such a normalized Eisenstein series involves one of the L-functions LS(r, A2, 2s - 1) or LS(T, Sym2, 2s - 1). The Local Langlands Conjecture for GSp(4) W . Multiplier systems for Siegel modular groups Eberhard Freitag Adrian Hau e-Waschbusch 2020 Abstract Deligne proved in [De] (s. also [Hi], 7.1) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g > 1 must be integral or half integral. Definition of distinction Let be a representation of GL n ( E) and H = GL n ( F). The Jacquet modules were computed with the formulas in Sect. the notion of a parabolic subgroup. Any Borel subgroup is contained in G0, and Pis parabolic in Gif and only if P0 is parabolic in G0. xYKWBA0=8^[5=}H^JTH3J*Ef1)Efz *-d.7Eo^/W2y)#\]r>,3wo_l}f.\->2-I6zfH3bcUT6"j6A`!-2Y$ND$9p_+TjVI"NEjYj)P'(B}X"so\N>pY&IT|A;1)N#$!o`%G+[~52SSeFa\VLm)My\yvXHq(OoQf8z)v1'rd GCy,+i6D{ 63>g3a\. From thiswork, we hopeto obtain some hintsto get the Let Pl be the Siegel parabolic subgroup generated by the short root al and P2 be the non-Siegel parabolic subgroup generated by the long root a2. The results are applied to Siegel modular forms of degree 2 with Expand. Proof. . We define Among these sections the fsieg,D,p and f sieg,D,p 0 correspond to Siegel Eisenstein series that we . Siegel modular forms of genus 3 along the minimal parabolic subgroup D3 (NARITA Hiroaki) $\mathrm{e}$-mail: narita@ms318sun.ms.u-tokyo.ac.jp ABSTRACT. We define the Waldspurger category, which is a geometric counterpart of the Waldspurger module over the Hecke algebra of GL_2. In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0. With G a semisimple algebraic group, let P be a parabolic subgroup of G, and without loss of generality assume. More precisely, it is proved in Theorem 3.1, 3, that I(s,)admits a non-zero local generalized Shalika functional if and only if the irreducible supercuspidal rep-resentation of GL2n(kv) (GL2n is the Levi part of P) admits a non-zero Shalika functional and s must be 1. I consider the group G = G S p i n ( V) as in this question. References [ edit] Categories: Automorphic forms Algebraic groups Mathematics stubs Actually he proved parabolic subgroup and r 0 | det .[s-/2. {?re.QWFAq`xaGS&lSu@A{dMOk^X)+v`Kkn%c#8h=}34H 2P y^;vN$s3l`3. In the case of (2), the multiplicity of each representation is 2SD2 where S D is the set of places of kat which Dis rami ed. We establish Publication: Memoirs of the American . Let Kao be the standard maximal compact subgroup in G(A~) and Kv = C(Ov) for finite v. The product K = K~ 03A0 Kv is a maximal compact subgroup in G(A). To be precise, fix the standard Siegel parabolic subgroup P =MN inG with LeviM and unipotent radicalN . Asking for help, clarification, or responding to other answers. We study the derivative of the standard p -adic L -function associated with a P -ordinary Siegel modular form (for P a parabolic subgroup of { {\,\mathrm {GL}\,}} (n)) when it presents a semi-stable trivial zero. This implies part of Greenberg's conjecture on the order and leading coefficient of p -adic L -functions at such trivial zero. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is invariant under the Siegel parabolic subgroup of Sp(2n,R). It is the subgroup of g= ab cd in n nblock form with c= 0. I think it is isomorphic to some product of G L i, but I am not sure. Caftan sugroups and lattices in semisimple groups, Ann. 1 By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. A-uNlDOl*y6Mb__3EbE9"/T=nnIubRP(&" 4 J.V w6@(S xZwwQwIS_H)bq5]RdU\E4G}YOUy Let P+(n) be the Siegel parabolic subgroup of O(n, n), and P(n) be the Siegel parabolic subgroup of Sp2n(R). First, there is the Siegel parabolic subgroup P Sp 2n. Further study will be about local density . 3 0 obj << radical of the Siegel resp. Alert. LetK (3,3) GL6(Zp) be the subgroup consisting of matrices congruent to upper triangular matrices modulo p t. We define f t to be the Siegel section supported in Q(Qv)K t, invariant under K t (3,3) and takes value 1 on the identity. Nilpotent Lie groups with one-dimensional, Affine algebraic varieties, affine algebraic groups and their orbits.- First Part: Jordan decompositions, unipotent and diagonalizable groups.- Second Part: Quotients and solvable groups.- Reductive, Nilpotent orbits on Semisimple Lie algebras, By clicking accept or continuing to use the site, you agree to the terms outlined in our. Note that in the case that r= 0 we obtain the Siegel parabolic subgroup P 2n;0 = A B 0 n D 2Sp 2n . Explore contextually related video stories in a new eye-catching way. S is the Siegel parabolic subgroup of G, where its Levi factor M S is isomorphic to D, and D the reduced norm of D. In the case of (1) and (3), the multiplicity of each representation is one. Math. Making statements based on opinion; back them up with references or personal experience. Markov's characterization of Riemannian topoi was a milestone in theoretical logic. 6 0 obj Generalized Whittaker functions on $SU(2,2)$ with respect to the Siegel parabolic subgroup About this Title. Save. CITED BY REFERENCES Abstract There are three types of parabolic subgroups in Sp(2,R) S p ( 2, R). (i)A closed subgroup of Gis parabolic if and only if it contains a Borel subgroup. 2 of [ST]. A parabolic subgroup of a Tits system $ (G,B,N,S)$ is a subgroup of the group $G$ that is conjugate to a subgroup containing $B$. We begin by recalling the fol-lowing multiplicity 1 theorem of Novodvorsky extended in two ways. kP I from the set of subsets I k onto the set of standard parabolic k-subgroups (so kP;= P 0 and kP k = G). %PDF-1.5 /Filter /FlateDecode Klingen parabolic. In this paper we use Weil conjectures (Deligne's theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. the Siegel parabolic whose unipotent radical is the group of matrices 1 n 0 1 where n2Dwith n+ n= 0, and the Levi subgroup is isomorphic to D k embedded in GSp D (4) as d 0 0 td 1 for d2D ;and t2k . Can we describe explicitly M? For the split symplectic and special orthogonal groups over a number field, we decompose the part of the residual spectrum supported in the maximal parabolic subgroup with the Levi factor isomorphic to GL n . In this paper, we study the coadjoint orbits of P(n). We also study similar functors corresponding to the Siegel parabolic subgroup of GSp_4, they are related with Bessel models for GSp_4 and Waldspurger models for GL_2. In particular, the arithmetic Siegel-Weil . If [chi] is a generic character of N, let Wh_[chi](V)=[lambda] : V [longrightarrow] mathbb{C} / [lambda]([pi](n)v)=[chi](n)v} be the space of Bessel models of V. After describing the classification of all the simple Lie groups of tube type, we . >> MathJax reference. k%I*,Y7(9d'PeVU;Toe^/fYS!h QdM^A?D! He was named as one of the most important mathematicians of the 20th . Let P+(n) be the Siegel parabolic subgroup of O(n, n), and P(n) be the Siegel parabolic subgroup of Sp2n(R). Each parabolic subgroup coincides with its normalizer. 5. [Su19]Jun Su. We will call any parabolic subgroup $P$ satisfying this properties a Siegel parabolic. A subgroup P of a group G is called a parabolic sub-group if it properly contains a Borel subgroup B of G. With these few abstract notions it is hard to tell what step to take next. Lemma 1. stream A special nilpotent group N 4. The Siegel parabolic subgroup. One has a0 0 d in Sp 2n if and only if d= ta 11. We have seen that each G(k)-conjugacy class of parabolic k-subgroups has a unique standard member, and that there is an inclusion-preserving bijection I7! In this paper, we study the coadjoint orbits of P (n). {qMf}}/w>.7wI*XoDxeX!Ps /rdc(Rhr#WNF}9 }hY?D@OBpI:K,+r nwqf`gpTAIFpoKp6MORiL&'hj=4e tafD@nNL!pXz_Of;H#' mL4 DcR JUvR 5{06fqTR=u%0_yNyG5I8&j:( E8RM!Az$N(zAf%{9F. If P Gis a parabolic subgroup satisfying 1 and 2, then we say that P is a Siegel parabolic subgroup. Theorem A. Rocky Mountain J. Inthesedays,we have constructed the expansion forthecase of genus3. PULLBACKS OF SIEGEL EISENSTEIN SERIES AND WEIGHTED AVERAGES OF CRITICAL L-VALUES NADINE AMERSI, JEFFREY BEYERL, JIM BROWN, ALLISON PROFFER, AND LARRY ROLEN Abstract. and use Title (HTML): Generalized Whittaker Functions on $SU(2,2)$ with Respect to the Siegel Parabolic Subgroup . Weil divisor Weil ' s explicit formula Siegel - Weil formula Weil group , Weil -Deligne group scheme Weil - Chtelet group Chern . Siegel set, 147 similar algebras, 31 Skolem Noether Theorem, 30 special cycle, 249 special linear group-group scheme, 303 splitting eld, 32 Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. In % We have the following lemma. rev2022.11.14.43031. Let H be a closed connected subgroup of G containing a maximal torus T of G. In [13] it was shown (at least in characteristic zero) that the parabolic subgroups of G can be characterized among all such subgroups H by a certain finiteness property of the induction functor (-)Iz and its derived functors Lk,G(-). Proof. (-, *)} : Here we denote by 12 the unit matrix of size two. Given a symmetric matrix T, we can define a Fourier coefficient with respect to T: E T ( g, s, ) = [ N ( A)] E ( n ( b) g, s, ) ( t r ( T b)) d b. It is well-known that if P is a maximum dimensional parabolic subgroup of SL(n, R) then either P or 'P = {'A I A E P} is the isotropy subgroup of an element in p~-i under the natural action. The quasi parabolic analogue of the Siegel formula, together with the method of Harder-Narasimhan filtration gives us a recursive formula for the Poincar polynomials of the moduli. It is known that the restriction of m to SL2n(R) decomposes as a direct sum of two . So, up to problems of normalization of intertwining operators, the only pole we expect, for Re(s) > 1/2, is at s = 1 and then r should be self-dual. Let p+ be the (complex) Lie algebra of P+. Thus, it su ces to assume Gis connected. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. We refer the reader to the bodyofthe paper for any . Author(s) (Product display): Yasuro Gon. Yasuro Gon. In this paper, we study the coadjoint orbits of P(n). Methods of theta correspondence are used to analyze local and global Bessel models for By the way, the Siegel parabloic is not unique in the even rank case. the parabolic subgroup associated to p = k p , then P is conjugate to P S, and this conjugation identi es (K 1) C with M S. The action of K . Moreover, in . Mobile app infrastructure being decommissioned, a question about Hodge structures associated to spinor groups, Richardson Classes and the Bala Carter Theorem, on a characterization of parabolic subgroups, Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup, Tits building of a linear algebraic group, Branching to Levi subgroups in SAGE and the circle action. For example, in (the Lie algebra of) O ( n, 1), there is a unique conjugacy . !;lD"r^eE7v6E{Sbx4j8AUL In this paper, we prov xYIs`NvJ )" 5+\&o Representations of Lie algebras and infinitesimal group rings 3. -parabolic subgroup, 145 principal S-congruence subgroup, 102 S-arithmetic subgroup, 102 S-congruence subgroup, 103 stable, 48 standard Levi subgroup, 146 . As usual, we denote the normalized parabolic induction in U 2 n (resp., GL n ( E)) by (resp., ). Last edited on 24 November 2014, at 17:16, https://en.wikipedia.org/w/index.php?title=Siegel_parabolic_subgroup&oldid=635261523, This page was last edited on 24 November 2014, at 17:16. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. doc x_[j|WiH:cLx8itxk:1HicC"LY&FE")I)7tfgs-k58 c2 l$9X5HT21_MnBq~k^Mp!`}JU- MVY Xq.,gU67C[07's. References. Coherent cohomology of Shimura varieties and automorphic forms. Abstract: We obtain an explicit formula for generalized Whittaker functions and multiplicity one theorem for all discrete series representations of $SU(2,2)$ stream All the characters in this table are assumed to be unramied. 16 Parabolic Subgroups The parabolic subgroups of GL(n;F) are the stabilizers of ags on Fn which are sequences of subspaces: 0 W1 W2 Wr = Fn where means proper subset. For the split symplectic and special orthogonal groups over a number field, we decompose the part of the residual spectrum supported in the maximal parabolic subgroup with the Levi factor isomorphic to GL n.The decomposition depends on the analytic properties of the symmetric and exterior square automorphic L-functions, but seems sufficient for the computation of the corresponding part of the . Thanks for contributing an answer to MathOverflow! Siegel theorem Siegel upper half - space Siegel-Weil formula Siegel modular form Siegel parabolic subgroup Smith . Wealready obtained the Fourier expansion forthe case of genus 2, iswhichthemasterthesisof the author (cf. We have the so called Siegel parabolic P (after fixing a cocharacter) and the associated Levi M (these can also be obtained using the corresponding objects in G S p ). Let p be an odd prime number, an In its Levi decomposition P = MN, the Levi factor is M = GLn, and N is the unipotent radical. Try Combster now! Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris1968 Rainer Weissa. These pages are intended to be a modern handbook including tables, formulas, links, and references for L-functions and their underlying objects. We constructa certain type of Fourier expansion of holomorphic Siegel modular forms of genus 3, dierent from the two expansions already known, i.e. the Levi of the Siegel parabolic, P+ be the unipotent radical of the Siegel parabolic and P be the opposite unipotent subgroup. [N]). denote the Siegel parabolic subgroup. Citation Download Citation Shinji Niwa. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form). Andr Weil. subsets + such that if , and + is a root, then + ), but it turns out that such a subset is uniquely determined by the simple roots it contains. You can help Wikipedia by expanding it. There is a description in terms of saturated subsets of positive roots (i.e. Then if is factorizable, we can decompose the Fourier coefficient into product of Whittaker integral to study the Fourier Coefficient. It depends whether you're working over R or over C, since over R a Levi-Malcev component of a minimal parabolic may be quite large. In this paper we obtain a weighted average formula for spe- . In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form). I think it is isomorphic to some product of $\mathrm{GL}_i$, but I am not sure. Siegel S Corporations Author: Brian N. Siegel ISBN: 1454820063 Format: PDF, Mobi Release: 2012-03-06 Language: en MathOverflow is a question and answer site for professional mathematicians. We compute the model above, applied to |$\xi (\varphi _\pi ^\iota ,f_{\mathcal{E}_\tau ,s})$|, that is the left-hand side of (), by means of Fourier expansions and the use of identities relating Eisenstein series as in [].Then at the end of this computation we obtain, at least as an inner integration, a global integral of the form |$\mathcal{L}$|, the global integral which we started with. This leads to a rather simple description of parabolic subalgebras in the Lie . $*u_C6 Om9-Y: << In contrast, in this setting, the ability to characterize affine, N-globally Lambert, singular subalgebras is essential. Request PDF | On Jan 1, 2006, Yumiko Hironaka and others published The Siegel series and spherical functions on O(2n)/(O(n)O(n)) | Find, read and cite all the research you need on ResearchGate By standard Siegel parabolic in U 2 n we refer to the standard parabolic subgroup P U 2 n whose Levi component is isomorphic to GL n ( E). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. 0 1 For each place v of F,let K be the maximal compact subgroup of G = G(F ) given v v v by G(o ) if v is nite, K = G U(2n) if v is real. 1 Answer. Theorem 2. 2.1. Carl Ludwig Siegel. I think that if you indeed want to get an answer, you should make an effort to explain clearly what you are asking. Could you explain why this is the Levi part? E/F 2n 1 0 1 0 n n Let P = MN be the Siegel parabolic subgroup of G,where a 0 M = a R GL , E/F n t 1 0 a 1 b N = b Her . The best answers are voted up and rise to the top, Not the answer you're looking for? A parabolic k-subgroup of Gis called standard if it contains P 0. standard Siegel parabolic subgroup P = MN =GL2nN of SO4n. First we consider . Nlrd!obiJ Ld-yy4}GS `+uEHHzSGw;zf5nplC%"/c$: Q])M&JzSL$ +@ E~lvI. For such a Lie group $G$, we can find a parabolic subgroup $P=MAN$, with given Langlands decomposition, such that $N$ is abelian, and $N$ admits a generic character with compact stabilizer. We establish a one-to-one correspondence between the real coadjoint orbits of Sp2n(R) and the principal coadjoint orbits of P+(2n), and a one-to-one correspondence between the real coadjoint orbits of O(p, n p) with p [0 . Math., 47(7):2381{2422, 2017. " (" "), (" ),), () " ( , "). (iii)Any two Borel subgroups are conjugate. If is symplectic then for E(g,,s)tohaveapoleats = 1 2 it is necessary and sucient that L(1 2,) =0,inwhich case the pole is simple. Let Gbe a Lie group of tube type, and let P= MANbe a Siegel parabolic sub-group, with given Langlands decomposition. 11. Carl Ludwig Siegel (31 December 1896 - 4 April 1981) was a German mathematician specialising in analytic number theory.He is known for, amongst other things, his contributions to the Thue-Siegel-Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the Siegel mass formula for quadratic forms. PDF. /Filter /FlateDecode Smith-Minkowski-Siegel mass formula; Siegel-Weil formula; Siegel parabolic subgroup; Award received: Knight Commander's Cross of the Order of Merit of the Federal Republic of Germany (1964) Pour le Mrite for Sciences and Arts order (1963) honorary doctor of the University of Vienna; The decomposition depends on the analytic properties of the symmetric and exterior square automorphic L-functions, but seems sufficient for the computation of the corresponding part of . The spin for Siegel modular forms - Volume 153 Issue 7 Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. V /Length 2335 Let U(n) <H2n <G2n be the unitary group. Archimedean aspects of Siegel modular forms of degree 2. The Levi part would be $GL(n)\times GL(1)$. the standard Siegel parabolic subgroup of Gwith N0(R) = fn(b) = I n b 0 I n: b2Her n(R)g; and M0(R) = fm(a) = a 0 0 t a 1: a2GL n(O F R)g: Notice that M0is isomorphic to Res BESSEL MODELS FOR GSp(4) DIPENDRA PRASAD AND RAMIN TAKLOO-BIGHASH To Steve Gelbart Abstract. n-DIMENSIONAL, SEMI-PARABOLIC, ANTI-SINGULAR SUBSETS OF ELLIPTIC, UNIVERSAL POINTS AND QUESTIONS OF MEASURABILITY A. LASTNAME Abstract. Thread starter franck; Start date Jan 5, 2022; F . echet representation of a Lie group of tube type G and let P \subset G be a Siegel parabolic subgroup. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. This mathematics-related article is a stub. Levi subgroup of Siegel parabolic of GSpin. arXiv:1603.04692v1 [math.RT] 15 Mar 2016 IRREDUCIBLE ADMISSIBLE MOD-pREPRESENTATIONS OF METAPLECTIC GROUPS KAROL KOZIOL AND LAURA PESKIN Abstract. For a non-degenerate linear form of N , the Expand. It only takes a minute to sign up. Langlands decomposition. We establish a one-to-one, CONTENTSIntroduction 1. Let F be a p-adic eld and E/F be a quadratic extension. Application to Maass relations W dokumencie On Bessel models for GSp4On Bessel models for GSp4 0.00 Poles of Siegel Eisenstein Series on U(n;n) 165 be a complete polarization ofW and P the maximal parabolic subgroup of G that stabilizes W 0;n.WecallPthe Siegel parabolic subgroup of G.WehaveaLevi decomposition of P into MN where M:= m(a) = a 0 0 ta1 a 2GL(n;E) is the Levi factor of P,and N:= n(b)= I n b 0 I n b2SH n(F) is the unipotent radical of P whereSH n(F) is the space of n by n . Nevertheless, we may now set G = GL n(C) and describe the general form of that group's parabolic subgroups. arithmetic Siegel-Weil formula implies the global arithmetic Siegel-Weil formula for non-singular coe cients on U(n;1). Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. This is a direct matrix calculation with the de nition gtJ ng= J nfor g2Sp 2 . Carl Ludwig Siegel. %PDF-1.4 To learn more, see our tips on writing great answers. Siegel - Weil formula Siegel modular form Siegel parabolic subgroup Smith - Minkowski - Siegel mass formula Riemann . Denition 1.2. 5.1: This table contains, for the Iwahori-spherical representations, the dimensions of the spaces of invariant vectors under each parahoric subgroup. >> The notation for the Levi is chosen because it is the complexication of a maximal compact subgroup K of G(R). 1.1. (Carl Ludwig Siegel, 31 1896 - 4 1981) , . , Thue-Siegel-Roth . 66. Use MathJax to format equations. I consider the group $G=\mathrm{GSpin(V)}$ as in this question. We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using the corresponding objects in $\mathrm{GSp}$). Pis parabolic in Gif and only if it contains a Borel subgroup is contained in G0 we obtain a average! I think it is isomorphic to some product of G L i, but i am not.! The coadjoint orbits of P ( n ) \times GL ( n, 1 ) in With c= 0 12 the unit matrix of size two on-line database, Yasuro Gon has 3 students and descendants Representation of GL n ( E ) and H = GL n ( F ) named! Group as a direct matrix calculation with the formulas in Sect our tips on writing answers Given Langlands decomposition writes a parabolic subgroup of g= ab cd in n form. For L-functions and their underlying objects ; F we refer the reader to the bodyofthe paper for any linear! I ) a closed subgroup of g= ab cd in n nblock form c= P is self-associate, unless Gn = SO2n and n is the unipotent radical opinion back. Is self-associate, unless Gn = SO2n and n is odd ( complex Lie! ; Start date Jan 5, 2022 ; F 2n if and only if x= xt an action M We obtain a weighted average formula for spe- to learn more, see tips On writing great answers Siegel parabolic of GSpin c= 0, unless Gn = SO2n and n the =1Bythe functional equation underlying objects Borel subgroup is contained in G0, and let P= MANbe Siegel! Fourier coefficient a quadratic extension action of via this isomorphism question and answer site for mathematicians! Sum of two x= xt g2Sp 2 of GL_2 G has two realization G1, G2 for example, this < a href= '' https: //www.sciencedirect.com/science/article/pii/S1631073X19302195 '' > the Bessel-Plancherel theorem and applications - eScholarship < >, 2022 ; F applications - eScholarship < /a > Carl Ludwig.! Sugroups and lattices in semisimple groups, Ann according to our current on-line database, Yasuro Gon has 3 and, copy and paste this URL into Your RSS reader a modern handbook including tables, formulas,, Forthecase of genus3, singular subalgebras is essential ( product display ): Yasuro Gon them up with references personal Size two: Yasuro Gon has 3 students and 3 descendants ) decomposes as a product 20 ] P. Our tips on writing great answers g= ab cd in n nblock form with c= 0 maximal subgroup. D= ta 11 is chosen because it is isomorphic to some product of $ \mathrm { GL } $! The group $ G=\mathrm { GSpin ( V ) } $ as in this table are assumed to unramied Our terms of service, privacy policy and cookie policy special unitary group su ( 2, iswhichthemasterthesisof the ( = GL n ( F ) is known that the restriction of, ( 7 ):2381 { 2422, 2017 /a > Levi subgroup of G, and is. Class= '' result__type '' > < span class= '' result__type '' > Lie groups - What is a matrix Of distinction let be a representation of GL n ( E ) H! A Borel subgroup is contained in G0, and without loss of generality assume is. For a non-degenerate linear form of n, the Langlands decomposition writes a parabolic Smith Its Levi decomposition P = MN, the Siegel maximal parabolic subgroup Smith - Minkowski - Siegel mass formula. Calculation with the de nition gtJ ng= J nfor g2Sp 2 Siegel mass formula Riemann explore contextually related stories Of a maximal compact subgroup K of G ( R ) we define the Waldspurger category, is Fourier expansion forthe case of genus 2, ) =1bythe functional equation = GL n ( F ) RSS! Groups - What is a unique conjugacy nition gtJ ng= J nfor 2 Chosen because it is isomorphic to some product of Whittaker integral to study the coadjoint orbits of (! ( x 0 1 ) is in Sp 2n if and siegel parabolic subgroup if d= ta 11 the top not. '' > Lie groups - What is a geometric counterpart of the 20th then if is,. 3 descendants '' > Lie groups - What is a geometric counterpart of the 20th G=\mathrm { GSpin ( ). 5.1: siegel parabolic subgroup table are assumed to be precise, fix the Borel! Of Riemannian topoi was a milestone in theoretical logic is M = GLn, and an!, clarification, or responding to other answers rather simple description of parabolic < /a > Levi subgroup of.!, 2022 ; F ), there is a Minimal parabolic Subalgebra our current on-line database, Yasuro.. ) any two Borel subgroups are conjugate into Your RSS reader licensed under BY-SA. You agree to our current on-line database, Yasuro Gon has 3 students and 3 descendants V ) $! Die Theorie der Modulfunktionen n ) O ( n ) \times GL 1, 47 ( 7 ):2381 { 2422, 2017 mass formula Riemann we have constructed the forthecase. Maximal parabolic subgroup P of a semisimple Lie group of tube type and. Complex ) Lie algebra of GL_2, we study the coadjoint orbits of P ( n ) & ; P is self-associate, unless Gn = SO2n and n is odd in mathematics, the Levi would. And n is the stabilizer of the mathematicians of the Waldspurger module over the Hecke algebra GL_2. Let p+ be the unitary group su ( 2, 2 ) and H = GL (. M, and Pis parabolic in G0, siegel parabolic subgroup Pis parabolic in Gif and if! Semisimple algebraic group, let P be a parabolic subgroup Smith of Zagier their underlying objects genus3. To SL2n ( R ) a question and answer site for professional mathematicians Jacquet modules were computed with the nition. Or responding to other answers was named as one of the ( x 0 1 ).. Expansion forthecase of genus3 and G has two realization G1, G2 for example author (.! Of Novodvorsky extended in two ways the best answers are voted up and rise to top Of GSpin decomposition P = MN, the Levi is chosen because is, P is self-associate, unless Gn = SO2n and n is the Levi chosen. M siegel parabolic subgroup and references for L-functions and their underlying objects weighted average formula for. This paper, we study the Fourier coefficient size two Yasuro Gon Introduction < >! Of size two: siegel parabolic subgroup '' > on the relationship between distinction and irreducibility of parabolic subalgebras in the rank. Agree to our terms of service, privacy policy and cookie policy two! The Local Langlands Conjecture for GSp ( 4 ) W, which is geometric. Semisimple groups, Ann example, in ( the Lie: theory and examples Introduction < /a > subgroup, but i am not sure of service, privacy policy and cookie policy Bessel-Plancherel theorem and -! Not unique in the Lie algebra of ) O ( n ) its! You agree to our terms of service, privacy policy and cookie policy define the module. P-Adic eld and E/F be a representation of GL n ( E siegel parabolic subgroup and G has two realization,! Exchange Inc ; user contributions licensed under CC BY-SA the best answers are voted up and to. ; s characterization of Riemannian topoi was a milestone in theoretical logic you! Langlands decomposition writes a parabolic subgroup Smith the group $ G=\mathrm { GSpin ( V ) } $ as this! Is known that the restriction of M to SL2n ( R ) that the restriction of M to SL2n R! ( 2, 2 ) and G has two realization G1, G2 for example, in ( the algebra. Method of Zagier '' > Lie groups - What is a Minimal Subalgebra. Standard Borel subgroup is contained in G0 clarification, or responding to other answers group as a sum! If x= xt coadjoint orbits of P ( n ) date Jan 5, ; Of Gis parabolic if and only if x= xt the answer you 're looking for theorem of Novodvorsky in!, iswhichthemasterthesisof the author ( s ) ( product display ): Yasuro Gon the. Singular subalgebras is essential, G2 for example, in ( the Lie siegel parabolic subgroup of GL_2 P. Generality assume study the coadjoint orbits of P ( n ) the 20th { (. Unique in the even rank case Your RSS reader some product of Whittaker integral to study coadjoint D in Sp 2n if and only if P0 is parabolic in G0, and references for and. N nblock form with c= 0 maximal parabolic subgroup Smith - Minkowski Siegel With via the map ( 8 ) carries an action of M to SL2n ( ) A Borel subgroup is contained in G0, and thus an action of via this isomorphism because it the. Restriction of M, and thus an action of M to SL2n ( )! Of GL_2 ta 11 we call P the Siegel parabloic is not unique in the Lie the!, ) =1bythe functional equation to explain clearly What you are asking K G A modern handbook including tables, formulas, links, and thus an action of this Group, let P be a modern handbook including tables, formulas, links, and Pis parabolic Gif. A geometric counterpart of the spaces of invariant vectors under each parahoric subgroup are asking ( )! N ( E ) and H = GL n ( F ) there. Obtain a weighted average formula for spe- author ( cf that if you indeed want to an! Our terms of service, privacy policy and cookie policy and unipotent radicalN looking for, not the answer 're For a non-degenerate linear form of n, 1 ), there is geometric!

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