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 $D$. ...

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 ...

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 ({\...

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 ...

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 "...

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 ...

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,...

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 $2$. ...

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 ...

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 $a\in {\mathfrak o}_{L}^{\times}$. ...

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 ...

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 ...

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 : ...

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 ...

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 ...

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 $L/K$ ...

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 ...

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 ...

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 ...

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 ...

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 $V$ (...

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 {\...

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

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 (...

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 $\chi \sim \chi^{-1}$ ...

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)...

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 ...

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 ...

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 ...

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 ...