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5
votes
1answer
130 views

Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic

I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field ...
4
votes
2answers
137 views

Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...
7
votes
2answers
407 views

On a theorem of Kazhdan

Let $G=GL_n(F)$, where $F$ is a p-adic local field, $U$ be the upper triangular maximal unipotent group, and $\theta$ a character of $U$. Then a Theorem of Kazhdan says that for any irreducible smooth ...
3
votes
0answers
46 views

Intertwining Operators Associated to Simple Reflections

Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi_s = Ind_B^G \chi\cdot ...
6
votes
1answer
140 views

Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about. Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' ...
6
votes
1answer
210 views

When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...
3
votes
2answers
505 views

orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$

Let $G\subseteq GL(n)$ be a linear algebraic group, and let $G({\Bbb Q}_p)\subseteq GL(V)$ act on a ${\Bbb Q}_p$-vector space V of finite dimension. Consider the action of $G$ on abelian subgroups ...
2
votes
2answers
107 views

admissible characters for $PGL_{2}(F)$

What are the irreducible admissible representations of $PGL_{2}(F)$ for $F$ a local nonarchimedean field and do we have formulas for their characters?
3
votes
0answers
82 views

Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
4
votes
3answers
359 views

What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise: Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...
1
vote
0answers
64 views

Are A and A^T unitarily equivalent over a p-adic field?

Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2. My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...
3
votes
1answer
183 views

Abelian varieties as analytic manifolds

Assume we have an Abelian varieties over the p-adic numbers, namely $ k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold. I am ...
2
votes
0answers
63 views

Mellin transform of Plancherel measure

Let $G$ be a reductive p-adic group with a chosen Haar measure $dg$. The Plancherel measure is the measure $\mu$ on the set of (tempered) irreducible representations of $G$ such that for any locally ...
3
votes
1answer
230 views

Abelian varieties over $p$-adic fields

Theorem : Let $A$ be an abelian varieties of dimension $d$ over a field $k$, non-archimedian valued complete, i.e. $\mathbb{Q}_p$, then $A(k)$ contains a subgroup of finite index analytically ...
4
votes
0answers
72 views

double coset decomposition of $U_\beta\setminus GL_n(E)/ U_\beta$ over p-adic field

Let $E/F$ be a quadratic extension of local fields. Let $\beta$ be a non degenerate Hermitian matrix of rank $n$, and let $U_\beta$ be the unitary group defined by $\beta$. Do we have an explicit ...
2
votes
1answer
139 views

Why are compactly induced representations projective in the category of admissible representations?

I am reading part of Dipendra Prasad's paper found here: http://arxiv.org/pdf/1306.2729v1.pdf. In it (in the middle of page 8) he writes that compactly induced representations are projective. Why is ...
1
vote
0answers
53 views

Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$. Let ...
3
votes
0answers
92 views

Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
2
votes
2answers
204 views

Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...
0
votes
0answers
56 views

Integrating new vectors of GL(n,F)

I'd be interested in the following: Let $\pi$ be an irreducible admissible generic representation of $GL(2n,F)$, $F$ a p-adic field. Assume that $\pi$ is ramified and let $W$ be a (non-trivial) new ...
0
votes
0answers
114 views

Thin profinite groups - nonabelian analogues of p-adic integers

Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...
1
vote
1answer
75 views

Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth ...
3
votes
3answers
191 views

Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...
7
votes
1answer
245 views

On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F. The following are equivalent (and groups satisfying these conditions are called unramified): (a) G is quasisplit ...
2
votes
0answers
84 views

A question on Plancherel measure for $p$-adic group

Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$. But at the same time, for example, in his famous ...
16
votes
2answers
676 views

Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of: Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...
1
vote
0answers
93 views

Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true? Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$. Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$. Let $U$ be ...
3
votes
1answer
84 views

Density of $\Gamma(N)$ in $\mathrm{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \not | N$

Let $\mathrm{Sp}_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$. ...
1
vote
1answer
123 views

$p$-adic analogues of $SO(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the ...
0
votes
0answers
40 views

Compatability of depth for elements in p-adic groups under base change

Suppose $F$ is a non-archimedean local field and $E$ is a tame extension of it. Let $G$ be any connected reductive group over $F$. For any $r>0$, let $G(E)_r$ be the set of elements with depth $\ge ...
4
votes
1answer
246 views

p-adic Lie group vs Lie algebra cohomology with mod p coefficients

My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$). Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action. Lazard ...
1
vote
1answer
181 views

Orbits of an action of maximal compact subgroups of p-adic orthogonal groups

Let $Q$ be a non-degenerate indefinite quadratic form on ${\mathbb R}^n$ and write $G=SO(Q)$ for the associated special orthogonal group. Let $K$ be a maximal compact subgroup of $G$ and consider the ...
5
votes
3answers
200 views

Questions on constructions of supercuspidal representations

To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, ...
1
vote
1answer
144 views

classification of irreducible finite dimensional representation of affine hecke algebra of type A

Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity. The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms ...
6
votes
2answers
158 views

Spherical functions for sl(2,Q_p)

I kindly would like to ask you the following- I am refering to page 175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms ..." My question to which I would kindly ask you to answer ...
4
votes
0answers
168 views

Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...
3
votes
0answers
92 views

Are the only discrete groups with nontrivial p-adic Haar measure finite?

Let K be a complete non-archimedean field (say $\mathbb{C_p}$) and let G be a discrete group. Since {e} is an open p-compatible compact subgroup of G, G admits a (left) K-valued Haar measure $\mu$. ...
2
votes
2answers
256 views

Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...
3
votes
1answer
176 views

Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center. Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$. Let $\chi : T(\mathbf{Q}_p) \to ...
1
vote
1answer
72 views

Strictly contracting elements in the center of a Levi subgroup

Let $G$ be a connected reductive group over a non archimedean local field $k$. Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...
1
vote
0answers
137 views

Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...
1
vote
0answers
158 views

Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field. Then $G(F)$ is a p-adic group. Let $\Psi(G)$ be the lattice of algebraic characters. Let $\Lambda_G$ be the ...
1
vote
0answers
67 views

Centralizer of a maximal split torus

Can you help me find a reference for the following fact? "If $G$ is a quasi-split $p$-adic group and $T$ is a maximal split torus in $G$, then the centralizer of $T$ is a maximal torus in $G$." Or ...
2
votes
2answers
212 views

fixed vector of a generic representation of GL(n,F)

Let $F$ be a locally compact non-archimedean field and $G_{n}$ the locally profinite group $GL(n,F)$. Let $\Gamma_{n,k}$ be the subgroup of $G_{n}$ whose elements are the matrices of the form $$ ...
6
votes
5answers
538 views

Is a unitary representation always semisimple?

I have been reading the online lecture notes by Fiona Murnaghan http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf The first lemma in p.35 says that every unitary representation of ...
5
votes
1answer
203 views

When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$. Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...
2
votes
1answer
334 views

On a result due to Zelevinskii

In his paper on the p-adic analogue of the Kazhdan-Lusztig hypothesis (Functional Analysis and Its Applications 15.2 (1981): 83-92), Zelevinskii proves a combinatorial proposition (outlined in Section ...
4
votes
0answers
113 views

parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...
2
votes
0answers
166 views

Correct definition of locally algebraic parabolic induction of a locally algebraic character

Let $L$ be a finite extension of $\mathbf{Q}_p$ and $G$ the group of $L$-points of a split connected reductive group $\mathbf{G}$ over $L$, $T$ the $L$-points of a split maximal torus in $\mathbf{G}$, ...
8
votes
2answers
395 views

Proving that some principal series representations of SL(2,F) are irreducible

I am sorry in advance if this question is not "research level". Let $F$ be a p-adic field. I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ ...