User paul broussous - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:33:39Z http://mathoverflow.net/feeds/user/4767 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130073/equivariant-and-compactly-supported-version-of-a-theorem-of-leray Equivariant and compactly supported version of a theorem of Leray Paul Broussous 2013-05-08T12:04:57Z 2013-05-08T12:04:57Z <p>In "Théorie des Faisceaux", Godement states the following theorem due to Leray (Theorem 5.2.5, page 209). </p> <p>Let ${\mathcal M}=(M_i )_{i\in I}$ be a locally finite closed covering of a topological space, and $\mathcal A$ a sheaf of abelian groups on $X$. Let $M$ be the nerve of the covering $\mathcal M$. Then there exists a spectral sequence whose $E_2$ term is $$E_2^{pq}=H^p (M, {\mathcal H}^q({\mathcal A}))$$ and whose $E_{\infty}$ term is the bigraduated group attached to a certain filtration of $H^* (X,{\mathcal A})$. </p> <p>Here the ${\mathcal H}^q({\mathcal A})$ are certain coefficient systems on the simplicial complex $M$ that Godement describes explicitely. </p> <p>In fact I would like to know whether there exists a version of this theorem in the case where there is a group $G$ acting on $X$, $\mathcal A$ is $G$-equivariant, and cohomology is replaced by cohomology with compact support. In fact the case where $\mathcal A$ is the constant sheaf with values in complex numbers would be sufficient. But I do need equivariance and compactly supported cohomology.</p> http://mathoverflow.net/questions/129081/steinberg-reps-of-reductive-groups-over-local-fields-vs-finite-fields/129205#129205 Answer by Paul Broussous for Steinberg reps of reductive groups over local fields vs finite fields Paul Broussous 2013-04-30T11:09:48Z 2013-04-30T11:09:48Z <p>For a general $G$, it is false that a general square integrable representation has a fixed non zero vector under the first congruence subgroup (even after a suitable twisting by a character) (there is a counter-example for e.g. ${\rm GL}(4)$). So in general you have to restrict to "level $0$" square integrable representations. For certain groups it is known that level zero square integrable irreducible representation are somehow parametrized by certain representation of $G(f)$. For exemple this is done by Silberger and Zink for the group ${\rm GL}(m,D)$, $D$ a division algebra:</p> <p>Silberger, Allan J.(1-CVLS); Zink, Ernst-Wilhelm(D-HUMB-IM) An explicit matching theorem for level zero discrete series of unit groups of $p$-adic simple algebras. (English summary) J. Reine Angew. Math. 585 (2005), </p> <p>P.S. For ${\rm GL}(p)$, $p$ prime, the non supercuspidal square integrable irreducible representations are up to twisting level $0$ representations. </p> http://mathoverflow.net/questions/129196/orbital-integrals-of-pseudo-coefficients-of-supercuspidal-reps/129203#129203 Answer by Paul Broussous for Orbital integrals of pseudo coefficients of supercuspidal reps Paul Broussous 2013-04-30T10:42:48Z 2013-04-30T10:56:32Z <p>In fact $\phi$ is not only a (multiple of a) pseudo coefficient, but is a (multiple of a) coefficient of $\pi$. See e.g. Carayol's article "Représentations cuspidales du groupe linéaire", Ann. ENS. </p> <p>Now to answer your main question, there is indeed a lot of such computations in e.g. the series of papers written by Bushnell and Henniart on explicit Jacquet-Langlands correspondence. For other references you may read the survey : </p> <p>Sally, Paul J., Jr.; Spice, Loren Character theory or reductive $p$-adic groups. Ottawa lectures on admissible representations of reductive $p$-adic groups, 103–111, Fields Inst. Monogr., 26, Amer. Math. Soc., Providence, RI, 2009. </p> <p>If youo are especially interested in supercuspidal representation of ${\rm GL}(2)$, you may read :</p> <p>Kutzko, Phil; Pantoja, José Character formulas for supercuspidal representations of the groups ${\rm GL}_2,\ {\rm SL}_2$. Comm. Algebra 26 (1998), no. 6, 1679–1697. </p> http://mathoverflow.net/questions/13526/geometric-interpretation-of-trace/125907#125907 Answer by Paul Broussous for Geometric Interpretation of Trace Paul Broussous 2013-03-29T12:01:37Z 2013-03-29T12:01:37Z <p>There is a special case where the trace has an obvious geometric interpretation. Assume that a group $G$ acts on a finite set $E$. It also acts on the vector space $F$ of functions on $E$ with values in some field $k$. Then if $g\in G$, the trace of the operator in ${\rm End}_k (F)$ attached to $g$ is the number of points in $E$ fixed by $g$. Very often in representation theory traces of operators are related to considerations on fixed point sets via Lefschetz type formulae.</p> http://mathoverflow.net/questions/115933/what-is-the-support-of-the-whittaker-function-of-a-new-vector-on-gl2/116053#116053 Answer by Paul Broussous for What is the support of the Whittaker function of a new vector on GL(2)? Paul Broussous 2012-12-11T07:34:47Z 2012-12-11T07:34:47Z <p>You should find all ingredients needed for your calculation in the following very useful notes by Ralf Schmidt and Mahdi Asgari :</p> <p>Some remarks on local newforms for GL(2). J. Ramanujan Math. Soc. 17 (2002), 115-147 </p> <p>It is available on line (Ralf Schmidt's webpage).</p> http://mathoverflow.net/questions/111564/what-is-the-plancherel-measure-for-textrmsl-3-mathbbq-p/111625#111625 Answer by Paul Broussous for What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$? Paul Broussous 2012-11-06T07:14:32Z 2012-11-06T07:14:32Z <p>This is not a complete answer but only some hints that could help you.</p> <p>Bushnell, Kutzko and Henniart have shown, for a general reductive group, that the restriction of the Plancherel measure to each block of the Bernstein decomposition may be computed via isomorphisms of Hecke algebras :</p> <p>Bushnell, Colin J.; Henniart, Guy; Kutzko, Philip C. Types and explicit Plancherel formulæ for reductive $p$-adic groups. On certain $L$-functions, 55–80, Clay Math. Proc., 13, Amer. Math. Soc., Providence, RI, 2011.</p> <p>In the following paper, Bushnell and Kutzko show that for certain blocks of ${\rm SL}(N)$the Hecke algebra is isomorphic to a Iwahori Hecke algebra for some ${\rm SL}(N')$ over another field. </p> <p>Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. I. Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 261–280. </p> http://mathoverflow.net/questions/109641/representations-of-reductive-groups-over-local-fields-through-parahoric-induction/109797#109797 Answer by Paul Broussous for Representations of reductive groups over local fields through parahoric induction Paul Broussous 2012-10-16T08:53:51Z 2012-10-16T14:03:59Z <p>This procedure allows to construct all "level $0$" irreducible representations of G. They appear as subquotients of your compactly induced representations. Here "level $0$ means that the representation has a non-zero fixed vector under the pro-unipotent subgroup of some parahoric. The answers to your questions are is the following paper:</p> <p>Morris, Lawrence Level zero $\bf G$-types. Compositio Math. 118 (1999), no. 2, 135–157</p> <p>If you use smaller groups $U_o \subset U$, then you can indeed get any irreducible representation as subquotient of a compactly induced representation. However when the compactly induced representation is irreducible it is automatically supercuspidal (see e.g. Bushnell-Henniart for a proof of that). All explicitely known supercuspidal representations are indeed obtained by compact induction. But it is still conjectural that they all are. </p> <p>In general a compactly induced representation from an irreducible representation of a compact open subgroup splits in two part. An admissible part which is a finite sum of supercuspidal representations and a non admissible part which contains non-supercuspidal as irreducible subquotients. </p> <p>To describe the non supercuspidal representations by compact open data, a good point of view is that of "types". You may read Bushnell and Kutzko's papers on that subject. </p> http://mathoverflow.net/questions/101067/are-all-irreducible-supercuspidal-representation-induced-from-compact-mod-center/101089#101089 Answer by Paul Broussous for Are all irreducible supercuspidal representation induced from compact-mod-center subgroups? Paul Broussous 2012-07-01T19:19:42Z 2012-07-01T19:19:42Z <p>It is known for GL(N) and SL(N) (Bushnell and Kutzko), for classical groups when the residue characteristic is not $2$ and when no quaternionic algebra is involved (Stevens), for GL(N) of a division algebra (Stevens and Sécherre), for a general reductive group when the residue characteristic of $F$ is large enough (Kim, Yu, ...). </p> http://mathoverflow.net/questions/96849/pseudo-coefficients-and-orbital-integrals/96881#96881 Answer by Paul Broussous for Pseudo coefficients and orbital integrals Paul Broussous 2012-05-14T10:02:39Z 2012-05-14T10:02:39Z <p>Existence of pseudo-coefficients for square-integrable representations (and the link with character values of the representations) is stated and proved in </p> <p>D. Kazhdan, Cuspidal geometry of $p$-adic groups. J. Analyse Math. 47 (1986), 1–36. </p> http://mathoverflow.net/questions/92227/about-the-maximal-abelian-subgroups-of-sl-2f/92232#92232 Answer by Paul Broussous for About the maximal abelian subgroups of $SL_2(F)$ Paul Broussous 2012-03-26T06:38:58Z 2012-03-26T06:42:52Z <p>If the characteristic of the field is not $2$, the description of non-split maximal tori is straightforward. Take an element $u$ of the base field $F$ which is not a square. Then you get such a torus as the set of matrices</p> <p>$$\left( \begin{array}{cc} x &amp; uy \newline y &amp; x \end{array} \right)$$</p> <p>where $x^2 -uy^2 =1$. Any maximal non-split torus is conjugate to this torus for some well chosen $u$.</p> http://mathoverflow.net/questions/91933/double-coset-decomposition-of-symplectic-group-over-a-quadratic-extension/91958#91958 Answer by Paul Broussous for Double coset decomposition of symplectic group over a quadratic extension Paul Broussous 2012-03-22T22:37:48Z 2012-03-22T22:37:48Z <p>This question is answered for a large number of reductive groups (including the symplectic) in a paper by P. Delorme and V. Sécherre (e.g. Math Arxiv, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces). </p> http://mathoverflow.net/questions/89184/representations-of-gl2-q-p-and-gl2-z-p/89254#89254 Answer by Paul Broussous for Representations of GL(2, Q_p) and GL(2, Z_p) Paul Broussous 2012-02-23T06:01:37Z 2012-02-23T06:01:37Z <p>Abbreviate $G={\rm GL}(n,F)$ and $K={\rm GL}(n,{\mathfrak o})$.</p> <p>The general philosophy of "type theory" is the following.</p> <p>When you restrict an irreducible representation of $G$ to $K$, you get a semisimple representation whose irreducible components are of two sorts:</p> <p>-- irreducible representations of $K$ occuring in "a lot" of representations of $G$ by restriction. </p> <p>-- the other.</p> <p>The first ones are uninteresting from the point of view of type theory. The second ones are called "typical" (say). </p> <p>To construct the smooth dual of $G$, Bushnell and Kutzko construct enough typical representations. But they are very far to have constructed the whole dual of $K$.</p> <p>So here are the answers to two of your questions :</p> <p>"Is the classification of the irreducible dual of $G$ known only modulo the representation theory $K$? NO</p> <p>"Do we know all the representations of $K$ needed for the dual of $G$? YES</p> http://mathoverflow.net/questions/89110/new-vectors-for-p-adic-groups/89133#89133 Answer by Paul Broussous for New vectors for $p$-adic groups Paul Broussous 2012-02-21T19:52:18Z 2012-02-21T19:52:18Z <p>The theory of new vectors for ${\rm GSp}(4)$ has been written by Schmidt and Roberts :</p> <p>Local Newforms for GSp(4). Springer Lecture Note in Mathematics, vol. 1918 (2007) </p> <p>See also Schmidt's webpage :</p> <p><a href="http://www2.math.ou.edu/~rschmidt/" rel="nofollow">http://www2.math.ou.edu/~rschmidt/</a></p> <p>The definition is trickier than in the case of ${\rm GL}(N)$</p> <p>By the way : there is a mistake in Jacquet/Piateski-Shapiro/Shalika. It was pointed out and fixed by Matringe :</p> <p>arXiv:1201.5506 Essential Whittaker functions for GL(n). Nadir Matringe. </p> <p>See also Jacquet's webpage.</p> <p>New vectors are also known for generic representations of reductive groups of small ranks (in fact of rank $1$) : ${\rm SL}(2)$, unitary groups.</p> <p>There is no general theory (except for spherical representations).</p> http://mathoverflow.net/questions/88991/exact-functor-and-representations-of-p-adic-groups/89002#89002 Answer by Paul Broussous for exact functor and representations of p-adic groups Paul Broussous 2012-02-20T08:58:54Z 2012-02-20T08:58:54Z <p>I do not think that your procedure gives the whole ${\mathcal R}({\mathbf G})$. In particular you'll have difficulties to get <em>irreducible representations</em> different from the Steinberg representation. However I do not know how to prove this fact! To see examples you may read my papers:</p> <p><em>Representations of ${\rm PGL}(2)$ of a local field and harmonic cochains on graphs</em>. Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 3, 495–513. </p> <p><em>Simplicial complexes lying equivariantly over the affine building of ${\rm GL}(N)$</em>. Math. Ann. 329 (2004), no. 3, 495–511. </p> <p>*Un revêtement de l'arbre de ${\rm GL}_2$ d'un corps local*. (French) [*A covering of the ${\rm GL}_2$ tree of a local field*] Compositio Math. 135 (2003), no. 1, 37–47.</p> <p>I construct certain simplicial complexes and study their cohomology.</p> <p>On the other hand, if you fix $X$ as being the Bruhat-Tits building and if you allow general equivariant coefficients (i.e. coefficient systems in the language of Schneider and Stuhler), then taking the (co)homology gives all smooth representations (maybe you have to add finiteness conditions here). This fact is proved in </p> <p>P. Schneider, U. Stuhler, * representation theory and sheaves on the Bruhat-Tits buildings*, Publ. Math. IHES.</p> http://mathoverflow.net/questions/88511/what-is-the-relation-of-the-absolute-galois-group-and-classical-profinite-groups/88517#88517 Answer by Paul Broussous for What is the relation of the absolute Galois group and classical profinite groups? Paul Broussous 2012-02-15T13:23:08Z 2012-02-15T13:50:55Z <p>This question has been solved by Paskunas in his PhD thesis:</p> <p><a href="http://front.math.ucdavis.edu/0306.5124" rel="nofollow">http://front.math.ucdavis.edu/0306.5124</a></p> <p>Corollary 8.2 of this reference gives an "inertial Galois correspondence" between supercuspidal types of the form $({\rm GL}(n,{\mathbb Z}_p), \lambda )$ and certain representations of the local inertial group. </p> <p>The case of non-supercuspidal types is a work in progress by Shaun Stevens and William Conley, certainly to appear somewhere soon.</p> http://mathoverflow.net/questions/86381/twisted-gelfand-pairs-reference-and-examples/86392#86392 Answer by Paul Broussous for Twisted Gelfand pairs (Reference and examples) Paul Broussous 2012-01-22T18:08:55Z 2012-01-22T18:08:55Z <p>These Hecke algebras are intensively studied in the field of "type theory" for reductive $p$-adic groups.</p> <p>You have a nice summary of basic facts <em>with proofs</em> in chapter 4 of Bushnell and Kutzko's book "The admissible dual of ${\rm GL}(N)$ via compact open subgroups" (the chapter is entitled "Interlude with Hecke algebras"). </p> <p>You may also read the monography "The Langlands conjecture for ${\rm GL}(2)$", written by Bushnell and Henniart. You'll find there a nice introduction to these algebras.</p> <p>There are many other references. But it depends on what exactly you're interested in.</p> http://mathoverflow.net/questions/84510/irreducibility-of-induced-representation-over-arbitrary-field/84519#84519 Answer by Paul Broussous for Irreducibility of Induced Representation over arbitrary field Paul Broussous 2011-12-29T14:39:18Z 2011-12-29T14:39:18Z <p>The inclusion ${\rm End}_H (V)\longrightarrow {\rm End}_G ({\rm Ind}_H^G V)$ is much clearer at the level of Hecke algebras. If ${\mathcal H}(G,V)$ denotes the spherical Hecke algebra attached to $V$ (which is known to be isomorphic to ${\rm End}_G ({\rm Ind}_H^G V)$), then ${\rm End}_H (V)$ naturally identifies with the subalgebra of functions in the Hecke algebra with support $H$ : to $\psi$ you attach the function with support $H$ given by $h\mapsto \pi (h)\circ \psi =\psi \circ \pi (h)$. Then one has just to observe that the hypothesis of Mackey's criterion exactly says that functions in ${\mathcal H}(G,V)$ have support in $H$ !</p> <p>For all of this you need to assume that the order of $G$ is prime to the caracteristics of the field of coefficients. </p> http://mathoverflow.net/questions/84448/analysis-in-division-rings/84516#84516 Answer by Paul Broussous for Analysis in Division Rings Paul Broussous 2011-12-29T14:09:15Z 2011-12-29T14:09:15Z <p>The classification of locally compact (non necessarily commutative) fields is done in chapter 1 of André Weil's Basic Number Theory. His method is very nice, based on the existence of a Haar measure on a locally compact group.</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84337#84337 Answer by Paul Broussous for 'Important' applications of p-adic numbers outside of algebra (and number theory). Paul Broussous 2011-12-26T20:44:51Z 2011-12-26T20:44:51Z <p>I am not convinced by the applications of $p$-adic numbers (or adèles) to theoretical physics, even though I am not a physicist. I think p-adic mathematical physics has so far nothing to do with real phenomena. But of course $p$-adic analysis is useful in mathematics. $p$-adic analysis has for instance natural applications in the $p$-adic Langlands program. The basic idea of that program is to replace the field $\mathbb C$ by a $p$-adic field when considering linear representations (of a $p$-adic Lie group or a Galois group). </p> <p>There are obvious applications of $p$-adic numbers and adèles to analytic number theory via the (classical) Langlands program. These applications are not only algebraic, since they may for instance predict the analytic behaviour of $L$-functions. </p> <p>Another interesting example is the existence of a nice locally compact topology, defined by Berkovich, on $p$-adic rigid manifolds (varieties over ${\mathbb C}_p$, the completion of the algebraic closure of ${\mathbb Q}_p$). You get in such a way varieties analogous to complex varieties. You can do very similar things like dynamics, dessins d'enfant, potential theory, integration of $1$-forms, ... The following survey articles by Ducros (for french readers) are excellent:</p> <p>Géométrie analytique $p$-adique : la théorie de Berkovich, Gazette des Mathématiciens 111 (2007), 12-27.</p> <p>Espaces analytiques $p$-adiques au sens de Berkovich, exposé 958 du Séminaire Bourbaki (mars 2006).</p> <p>This is a very promissing theory. </p> http://mathoverflow.net/questions/83694/normalizers-of-maximal-compact-groups/83736#83736 Answer by Paul Broussous for Normalizers of maximal compact groups? Paul Broussous 2011-12-17T18:42:56Z 2011-12-17T18:42:56Z <p>Hints :</p> <p>-- For $K= {\rm GL}(n,{\mathbb Z}_p )$. Make $G={\rm GL}(n,{\mathbb Q}_p )$ acts on ${\mathbb Z}_p$-lattices of ${\mathbb Q}_p^n$. Prove that the lattices stabilized by $K$ are the $p^k {\mathbb Z}_p^n$, $k\in {\mathbb Z}$. Observe that the normalizer $\tilde K$ of $K$ permutes these lattices and conclude that ${\tilde K}={\mathbb Q}_p^\times K$. </p> <p>-- For $K=O(n)$ (or $U(n)$). Do someting similar by making $G$ act on the set of positive definite symmetric (hermitian) matrices via $(X,A)\mapsto XA{\bar X}^{t}$. </p> <p>In general, for a non-archimedean base field, you can make $G$ act on the extended Bruhat-Tits building, but the answer is going to be technical according to whether $G$ has a center or not, is simply connected or not. Over archimedean fields, I guess you have to use symmetric spaces.</p> http://mathoverflow.net/questions/83695/what-is-the-normalization-factor-for-gl-n-mathbbq-p-gl-n-mathbb-mathb/83724#83724 Answer by Paul Broussous for What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$? Paul Broussous 2011-12-17T17:17:43Z 2011-12-17T17:17:43Z <p>The measure of $KxK$ is a classical computation that may be found in: Macdonald "Symmetric Functions and Hall Polynomials" (Oxford Mathematical Monographs), more precisely in Chapter V: The Hecke ring of ${\rm GL}(n)$ over a local field. </p> http://mathoverflow.net/questions/83419/what-is-a-canonical-set-of-representatives-in-gln-f-for-the-vertices-in-the-b/83464#83464 Answer by Paul Broussous for What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? Paul Broussous 2011-12-14T20:13:20Z 2011-12-14T20:13:20Z <p>You somehow want to parametrise the vertices of the building of $G={\rm GL}(n,F)$ : $$G/F^\times K = BK/F^\times K= B(F)/B({\mathfrak o})Z(F)$$ (by Iwasawa decomposition).</p> <p>For $n=2$ ou can easily find representatives, but for $n>2$, it's going to be tricky!</p> <p>I just give some hints. Write $N$ for the unipotent radical of $B$ and $T$ for the diagonal torus so that $B=T\ltimes N$.</p> <p>-- If $n,n'\in N$ and $t, t'\in T$, then if $nt\sim nt'$ mod $B(O)Z(F)$, one has $t\sim t'$ mod $Z(F)T(O)$. So you may assume that $t$ is of the form</p> <p>$$t= {\rm diag}(\varpi^{k_1}, ...,\varpi^{k_n})$$ where $(k_1 ,...,k_n )$ is well defined modulo the diagonal action of $\mathbb Z$ on ${\mathbb Z}^n$. </p> <p>-- You have $nt \sim n't$ mod $Z(F)B(O)$ iff $n\sim n'$ mod $tN(0)t^{-1}$.</p> <p>So for each $t$ as above, you need to find a system of representatives of $$N(F)/tN(O)t^{-1}$$ For $n=2$, this is easy. For $n>2$, this seems tricky. I've never tried ...</p> http://mathoverflow.net/questions/82196/elliptic-orbital-integral/82308#82308 Answer by Paul Broussous for Elliptic orbital integral Paul Broussous 2011-11-30T20:09:18Z 2011-11-30T20:09:18Z <p>You have introductions to the building of ${\rm GL}(n)$ in Brown's book "Buildings", or in Paul Garrett's book. The extended building is just the product of the non extended building $X$ with the real line $\mathbb R$, with the action $$g.(x,r)=(\ g.x \ , \ r+val_F (det(g)) )$$ It is the geometric realization of a simplicial complex whose vertex set is in equivariant bijection with the set of lattice in $F^n$. In the non extended building the stabilizer of a vertex is conjugate to $F^{\times} {\rm GL}(n,{\mathfrak o})$. In the extended building the stabilizer of a vertex is conjugate to ${\rm GL}(n,{\mathfrak o})$. </p> <p>When $\phi$ is the characteristic function of ${\rm GL}(n,{\mathfrak o})$, the value of the orbital integral is related to the number of lattices in $F^n$ fixed by $\gamma$, so to a set of fixed vertices in the extended building.</p> <p>Examples of calculations of orbital integrals using the building may be found:</p> <p>-- in the PhD thesis of J. Rogawski that you can find online : <a href="http://www.math.ucla.edu/~jonr/eprints.html" rel="nofollow">http://www.math.ucla.edu/~jonr/eprints.html</a></p> <p>-- in Langlands's book "base change for ${\rm GL}(2)$".</p> <p>-- in Kottwitz's PhD thesis : "Orbital integrals on ${\rm GL}_{3}$" Amer. J. Math. 102 (1980), no. 2, 327–384. </p> http://mathoverflow.net/questions/82196/elliptic-orbital-integral/82201#82201 Answer by Paul Broussous for Elliptic orbital integral Paul Broussous 2011-11-29T17:24:05Z 2011-11-29T17:24:05Z <p>The coset space $G/{\rm GL}(n,{\mathfrak o}_F)$ may be seen as the set of vertices of the extended Bruhat-Tits building of ${\rm GL}(n,F)$. There is a second point of view that amounts to consider this set as the set of $k_F$-rational points of a variety defined over $k_F$ (the residue field of $F$) : an affine flag variety. This is the point of view used by Ngo Bao Chau to prove the Fundamental Lemma.</p> http://mathoverflow.net/questions/82007/how-to-understand-the-representation-theory-of-sln-from-gln/82013#82013 Answer by Paul Broussous for How to understand the representation theory of $SL(n)$ from $GL(n)$? Paul Broussous 2011-11-27T16:37:38Z 2011-11-27T16:37:38Z <p>In the case of a non-Archimedean local field $F$, one may reduce the representation theory of $H={\rm SL}(n)$ to that of $G={\rm GL}(n)$. For instance supercuspidal representations of $H$ are obtained as constituents of the restriction to $H$ of the supercuspidal representations of $G$ (these restrictions are semisimple). In fact restriction of representations from $G$ to $H$ is an instance of Langlands functorialities. It corresponds to the natural projection between $L$-groups :</p> <p>$${}^{\rm L}G ={\rm GL}(n,{\mathbb C})\longrightarrow {}^{\rm L}H={\rm PGL}(n,{\mathbb C})$$</p> <p>Representations of ${\rm PSL}(n)$ are just the representations of $H$ with trivial central character.</p> <p>A good reference is :</p> <p>MR1253507 (94k:22035) Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. II. Proc. London Math. Soc. (3) 68 (1994), no. 2, 317–379. </p> <p>MR1209709 (94a:22033) Bushnell, Colin J.; Kutzko, Philip C. The admissible dual of ${\rm SL}(N)$. I. Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 261–280. </p> http://mathoverflow.net/questions/79809/iwasawa-decomposition-and-cartan-decomposition/79816#79816 Answer by Paul Broussous for Iwasawa decomposition and Cartan decomposition Paul Broussous 2011-11-02T10:13:44Z 2011-11-02T10:13:44Z <p>You have a proof which uses the fact that maximal compact open subgroups correspond to vertices in the building of ${\rm GL}(n)$. For instance for the Cartan decomposition you want to classify the orbits of $K={\rm GL}(n,{\mathfrak o}_F)$ (${\mathfrak o}_F$ denotes the ring of integers of your p-adic field $F$) in the vertex set of the building. To the aim, you first use the fact that $K$ acts transitively on the apartments containing the vertex fixed by $K$. This way you may send any pair of vertices $(s,t)$ in a fixed apartment $A$. The stabilizer of $s$ acts transitively on the Weyl chambers and you may assume that $t$ lies in a fixed Weyl chamber. From this it's easy to prove that $$t={\rm Diag}(\varpi_F^{k_1},...,\varpi_F^{k_n}).s$$ where e.g. $k_1 \geq k_2 \geq ...\geq k_n$ and where $\varpi_F$ is a uniformizer of $F$.</p> <p>Similarly the Iwasawa decomposition $G=KB$ corresponds to the fact that $K$ acts transitively on the germs of "quartiers" (you may prove this using the fondamental properties of the building stated in Bruhat-Tits, IHES, volume 1).</p> <p>In fact doing the proofs this way is cheating ! For Bruhat and Tits prove the basic properties of the building by using properties of the valuated root datum, that is by doing row and column operations on matrices ...</p> <p>But even if these proofs are not geniune "proofs", they allow you to "visualize" why these decompositions hold.</p> http://mathoverflow.net/questions/78312/centralizer-of-elliptic-elements-in-gl2/78316#78316 Answer by Paul Broussous for Centralizer of elliptic elements in $GL(2)$ Paul Broussous 2011-10-17T08:11:19Z 2011-10-17T08:11:19Z <p>The centralizer of $\gamma$ is a torus of the form $E^\times$, where $E/F$ is a quadratic field extension. Assume moreover that $\gamma$ is regular (so that $E/F$ is separable). The centralizer of $\gamma$ in $G_v$ is $(E\otimes_F F_v )^\times$. The algebra $E\otimes F_v$ is either a field (in that case $\gamma$ is elliptic in $G_v)$) or a sum of two fields isomorphic to $F_v$, according to whether the prime $v$ splits in $E$ or not. In the latter case $\gamma$ is a regular element lying in a split torus of $G_v$. </p> http://mathoverflow.net/questions/77829/drinfelds-coverings-of-p-adic-symmetric-domains-translated/77839#77839 Answer by Paul Broussous for Drinfeld's "Coverings of p-adic symmetric domains" translated? Paul Broussous 2011-10-11T16:45:15Z 2011-10-11T16:45:15Z <p>In the following paper (in French), the authors consider the case of the $p$-adic upper half plane (the case of dimension $2$) and construct its coverings following Drinfeld.</p> <p>Boutot, J.-F. and Carayol, H. Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfelʹd. (French. English summary) [$p$-adic uniformization of Shimura curves: the theorems of Cherednik and Drinfelʹd] Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque No. 196-197 (1991), 7, 45–158 (1992). </p> http://mathoverflow.net/questions/75571/iwahori-for-pgl-2/75578#75578 Answer by Paul Broussous for Iwahori for PGL_2 Paul Broussous 2011-09-16T07:11:44Z 2011-09-16T07:11:44Z <p>The (standard) Iwahori $\bar I$ of ${\rm PGL}(N,F)$ is the image of the Iwahori subgroup $I$ of ${\rm GL}(N,F)$ formed of the matrices with integer coefficients, upper triangular mod ${\mathfrak p}_F$, and with invertible determinant. To extract its defition from Bruhat-Tits general theory, there is an extra difficulty here coming from the fact that the group is not simply connected (contrary to e.g. ${\rm SL}(N,F)$). </p> http://mathoverflow.net/questions/70210/to-what-extent-do-we-know-the-representations-of-gl2-zp/70306#70306 Answer by Paul Broussous for To what extent do we know the representations of GL(2,Zp) Paul Broussous 2011-07-14T08:52:35Z 2011-07-23T20:02:12Z <p>All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski :</p> <p>Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$.</p> <p>Reference in Math. Arxiv : <a href="http://arxiv.org/abs/0807.4684" rel="nofollow">http://arxiv.org/abs/0807.4684</a></p> <p>But he does not follow the procedure that you propose at all. He uses much more generalizable tools : Clifford theory and an adapted version of Kirillov's orbit method. These ideas are now classical in type theory for $p$-adic reductive groups (see e.g. Howe and Kutzko's works). </p> <p>Some people are now working on the more general question of constructing the representations of ${\rm GL}(n,{\mathcal O})$ (A. Stasinski, A.--M. Aubert, F. Courtès, and others). This problem is known to be "wild"</p> http://mathoverflow.net/questions/129196/orbital-integrals-of-pseudo-coefficients-of-supercuspidal-reps/129203#129203 Comment by Paul Broussous Paul Broussous 2013-04-30T11:24:24Z 2013-04-30T11:24:24Z No the computations are done. http://mathoverflow.net/questions/126780/the-sum-of-same-powers-of-all-matrices-modulo-p Comment by Paul Broussous Paul Broussous 2013-04-10T05:59:02Z 2013-04-10T05:59:02Z One may ask the following question as well. Let $\Omega$ be an adjoint orbit of ${\rm GL(p,{\mathbb F}_p)$ in ${\rm M}(p,{\mathbb F}_p)$ and $k$ be a positive integer. Then $$\sum_{A\in \Omega} A^k$$ is a scalar matrix $\lambda (\Omega )I_p$. Can one find the scalar $\lambda (\Omega )$ ? http://mathoverflow.net/questions/101067/are-all-irreducible-supercuspidal-representation-induced-from-compact-mod-center/101089#101089 Comment by Paul Broussous Paul Broussous 2012-07-03T05:23:14Z 2012-07-03T05:23:14Z No, in all statements G is simply assumed to be defined over a local field. http://mathoverflow.net/questions/101067/are-all-irreducible-supercuspidal-representation-induced-from-compact-mod-center/101110#101110 Comment by Paul Broussous Paul Broussous 2012-07-03T05:21:03Z 2012-07-03T05:21:03Z That would be great indeed ! http://mathoverflow.net/questions/96849/pseudo-coefficients-and-orbital-integrals/96881#96881 Comment by Paul Broussous Paul Broussous 2012-05-14T20:27:43Z 2012-05-14T20:27:43Z I find Kazhdan's paper difficult to read ! http://mathoverflow.net/questions/96849/pseudo-coefficients-and-orbital-integrals/96881#96881 Comment by Paul Broussous Paul Broussous 2012-05-14T13:32:24Z 2012-05-14T13:32:24Z You also have the exact statement on the link bewteen orbital integral of a pseudo-coefficient and the character of the representation in Kazhdan's paper. http://mathoverflow.net/questions/89928/induced-character-for-non-injective-homomorphisms Comment by Paul Broussous Paul Broussous 2012-03-01T11:47:26Z 2012-03-01T11:47:26Z Isn't it enough to replace $H$ bu $\phi (H)$ to reduce to the injective case ? http://mathoverflow.net/questions/89669/theorem-of-cantor-bernstein-in-the-category-of-smooth-representation-of-g/89730#89730 Comment by Paul Broussous Paul Broussous 2012-02-28T07:55:58Z 2012-02-28T07:55:58Z More precisely, this comes from the fact that (since the center is compact) supercuspidal representations are projective and injective objects of the category of smooth representations. http://mathoverflow.net/questions/88991/exact-functor-and-representations-of-p-adic-groups Comment by Paul Broussous Paul Broussous 2012-02-20T08:40:39Z 2012-02-20T08:40:39Z How do you define exactness in the category of $G$-simplicial complexes ? I don't think this makes sense. http://mathoverflow.net/questions/88542/character-determines-the-representation/88558#88558 Comment by Paul Broussous Paul Broussous 2012-02-17T13:19:17Z 2012-02-17T13:19:17Z In the $p$-adic setting, as Anadimadhyanta wrote, the answer is yes for admissible complex representations of locally profinite topological groups. Another reference is Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a local non-Archimedean field. Uspekhi Mat. Nauk 10, No.3, 5-70 (1976), corollary (2.20) page 21. http://mathoverflow.net/questions/87333/stratum-corresponding-to-a-multiplicative-character-of-a-local-field Comment by Paul Broussous Paul Broussous 2012-02-02T20:15:39Z 2012-02-02T20:15:39Z Are you assuming that 2 is a uniformizer of your field F ? Things are definitely not clear ... http://mathoverflow.net/questions/87333/stratum-corresponding-to-a-multiplicative-character-of-a-local-field Comment by Paul Broussous Paul Broussous 2012-02-02T14:48:57Z 2012-02-02T14:48:57Z Do you mean : the theory of strata as developped by Kutzko and Bushnell ? What do you mean by &quot;v(2) = r&quot; ?! Could explain your notation ? Thanks http://mathoverflow.net/questions/86381/twisted-gelfand-pairs-reference-and-examples/86392#86392 Comment by Paul Broussous Paul Broussous 2012-01-23T19:52:53Z 2012-01-23T19:52:53Z I'm not a specialist of real groups, unfortunately. http://mathoverflow.net/questions/83419/what-is-a-canonical-set-of-representatives-in-gln-f-for-the-vertices-in-the-b/83464#83464 Comment by Paul Broussous Paul Broussous 2011-12-14T20:14:31Z 2011-12-14T20:14:31Z Here $O={\mathfrak o}$ is the ring of integers, and $\varpi$ a uniformizer. http://mathoverflow.net/questions/82196/elliptic-orbital-integral Comment by Paul Broussous Paul Broussous 2011-11-29T17:18:47Z 2011-11-29T17:18:47Z I guess that $x=\gamma$. I think your first question is not precise enough. What type of function $\phi$ are you interested in ? If $\phi$ is e.g. the characteristic function of ${\rm GL}({\mathfrak o}_F )$, then you have computations of the integral made by Langlands, Kottwitz, Rogawsky in very particular cases. I don't think that one can expect a nice general formula. But I may be wrong.