Paul Broussous
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Example for column rank $\neq$ row rank
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33 votes

Let $D$ be a skew field and consider the sets of $2\times 1$-matrices (columns) and $1\times 2$-matrices (lines) as left vector spaces over $D$. Let $a$ and $b$ be two non-commuting elements of $D$. ...

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Geometric interpretation of trace
17 votes

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 ...

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Is the Steinberg representation always irreducible?
14 votes

Consider the case where $\mathbb F$ is a non-achimedean local field. Then following Borel and Serre (Cohomologie d'immeubles et de groupes $S$-arithmétiques, Topology, 1976), one may equip $T_n ({\...

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Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?
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11 votes

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 ...

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Are all cuspidals induced?
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9 votes

Question 1. Yes indeed. a) There are new results for classical groups and their inner forms (works of Shaun Stevens, Daniel Skodlerack, ...). In particular Skodlerack proved that in the case of "...

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Why are spherical representations subquotients of unramified principal series?
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9 votes

Statement 2. comes from the following classical fact whose proof can be found in e.g. Bushnell and Kutzko, "The admissible dual of ${\rm GL}(N)$ via compact open subgroups". This is a particular case ...

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Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$
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9 votes

The answer to both questions is yes. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan, Representations of the group ${\rm GL}(n,...

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Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?
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9 votes

The symmetric space ${\rm GL}(2,{\mathbb F}_q)/T$, where $T$ is the diagonal torus, is not a Gelfand pair : the Steinberg representation contains the trivial character of $T$ with multiplicity $2$. ...

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What is the relation of the absolute Galois group and classical profinite groups?
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8 votes

This question has been solved by Paskunas in his PhD thesis: http://front.math.ucdavis.edu/0306.5124 Corollary 8.2 of this reference gives an "inertial Galois correspondence" between supercuspidal ...

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Is there any approximated version of Hilbert 90?
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8 votes

Yes such approximated versions of Hilbert 90 do exist. But you need some technical conditions. For instance assume that $L/K$ is unramified of degree $d$ and that $a\in {\mathfrak o}_{L}^{\times}$. ...

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On $XX'=I$ such that $AX=XB$ is true when $A,B\in\{0,1\}^{n\times n}$
7 votes

You question is easily seen to be equivalent to the following : is it true that two unoriented finite graphs are isomorphic if and only if they are isospectral ? Unfortunately the answer is no and ...

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Is the restriction of a representation semisimple?
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7 votes

The anwser to your question is "no in general" since you already have a counter-example in the case $n=2$. Take for $\pi$ an irreducible supercuspidal representation of ${\rm GL}(2,F)$. Then its ...

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New vectors for $p$-adic groups
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7 votes

The theory of new vectors for ${\rm GSp}(4)$ has been written by Schmidt and Roberts : Local Newforms for GSp(4). Springer Lecture Note in Mathematics, vol. 1918 (2007) See also Schmidt's webpage : ...

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Systems of imprimitivity for unitary representations - reference request
6 votes

This is not an answer, but here is another proof in the same spirit as yours. Write $(-\vert -)$ for the canonical scalar product of ${\mathbb C}^d$. Since the $G$-representation ${\mathbb C}^d$ is ...

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Do tori in a symplectic group always have invariant maximal isotropic subspaces?
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6 votes

When $K$ is finite or $p$-adic the answer to your question is negative. Indeed there exist maximal tori which are anisotropic. Those tori are not included in any proper parabolic subgroup, so they ...

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Surjectivity of trace map
6 votes

In addition to Jason's counter-example, we have the case of wild extensions of local fields (cf. Algebraic Number Theory, Cassels and Fröhlich ed., Chap. I "Local fields", by A. Fröhlich). Let $L/K$ ...

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

This follows from Frobenius reciprocity for compact induction. Let $G$ be the group of rational points of a reductive $p$-adic group and $K$ be a compact open subgroup of $G$. Then if $\lambda$ is any ...

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Maximal compact subgroup of p-adic lie groups
6 votes

If G is a simply connected semisimple group (e.g. ${\rm SL}(N)$, ${\rm Sp}_N$, ${\rm Spin_N}$, ...), then it is a theorem of Bruhat and Tits that there are exactly $l+1$ conjugacy classes of maximal ...

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Steinberg reps of reductive groups over local fields vs finite fields
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6 votes

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 ...

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Representations of GL(2, Q_p) and GL(2, Z_p)
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6 votes

Abbreviate $G={\rm GL}(n,F)$ and $K={\rm GL}(n,{\mathfrak o})$. The general philosophy of "type theory" is the following. When you restrict an irreducible representation of $G$ to $K$, you get a ...

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Irreducibility of Induced Representation over arbitrary field
6 votes

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 $V$ (...

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Normalizers of maximal compact groups?
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6 votes

Hints : -- 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 {\...

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To what extent do we know the representations of GL(2,Zp)
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6 votes

All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski : Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$. ...

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$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$
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5 votes

For two smooth representations $\pi_i$, $i=1,2$, of $G$, one has $\mathrm{Hom}_G (\pi_1 ,\pi_2 ) \simeq \mathrm{Hom}_G (\pi_2^\vee ,\pi_1^\vee )$. On the other hand the contragredient of $C_c^\infty (...

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Contragredient of a cuspidal representation
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5 votes

This is already false for $G={\rm GL}(1,F)$. In that case a cuspidal irreducible representation is a smooth character $\chi$ of $F$. The contragredient is $\chi^{-1}$. We have $\chi \sim \chi^{-1}$ ...

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Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
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5 votes

The answer to your questions (with proofs) may be found in C.J. Bushnell, P.C. Kutzko, The admissible dual of SL(N). I Annales scientifiques de l'École Normale Supérieure, Série 4 : Volume 26 (1993)...

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Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
5 votes

computing the trace of a smooth irreducible representation is a very difficult problem which is far from being totally solved. For a nice overview, you may read : Sally, Paul J., Jr.; Spice, Loren ...

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Definition of Hecke operators
5 votes

In order to understand the general philosophy of Hecke algebras, I recommand to do exercice 22 of Bourbaki "Groupes et Algèbres de Lie", Chap. IV. In this exercice a Hecke algebra is defined in ...

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Representation-theoretic operations on modular forms
5 votes

With the notation of the question, the tensor product $\rho_a \otimes\rho_b$ is a degree $4$ Galois representation so corresponds to an automorphic representation for ${\rm GL}(4)$. This ...

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Representations of reductive groups over local fields through parahoric induction
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5 votes

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 ...

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