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I answer the question in the title: $T$ can act trivial iff the rep is trivial. Here is the proof: We have $$ SL_2(Z/N) = \oplus_{p^k || N} SL(Z/p^k)$$ by the Chinese remainder theorem and because S …

I'll give some fairly general remarks - too long for a comment. I assume $G$ is reductive and $P$ parabolic. The Mackey Restriction Induction formula yields $$ Res_{H} Ind_{P}^G \pi = \int\limits_{\g …

Yes, the central character of the even $K$-types is trivial, and of the odd ones is the sign character. Also, this can easily be seen from the classification of irreducible representation on Hilbert …

Here are some observations, too long for a comment: 1) Note that cuspidal irreducible representation are compactly induced $\sigma = c-ind_K^G \tau = Ind_K^G \tau$ 2) You have the second adjointnes …

Given a locally compact group $G$ with a compact subgroup $K$. Assume we are given two irreducible, infinite dimensional, admissible representations $\pi$ and $\pi'$ of $G$. What are examples, w …

Is there a formula for the determinant of an induced representation, e.g. in the fashion of the Frobenius character formula. I would hope for something: $$ det \; Ind_H^G \rho(g) = (-1)^\alpha \p …

To my knowledge, it is not known whether $Ind_{P_{n,k}}^{GL_n(o)} 1$ decomposes with single multiplicity. This is certainly necessary by Frobenius reciprocity $$ dim Hom_{P_{n,k}}( 1 , Res_{P_{n,k}} …

I found Traces of Hecke operators by Knightly and Li very readable. They treat Gl(2,R) by a similar method. Knapp or Wallach is also nice to read and more general. They have chapters for Sl(2,R) and …

No, they are not. Consider $\rho_1$ and $\rho_1 \oplus \rho_2$. It the $\rho_j$ are pairwise disjoint, i.e., $Hom_G(\rho_j , \rho_l)= \{0 \}$ for $j \neq l$, then this is sufficient. This can be expre …

Because only infinite dimensional, unitary representation of $SL(2,F)$ for a local field $F$ can fail to be tempered, if they are spherical. This follows from the classification. This is true for $GL( …

Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi_j \rightarrow \pi$ if and only if $tr\; \pi_j \righ …

Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$. G …

If an irreducible representation is unitarizable, it has at most one unitarization (up to equivalence). Going back from $S_2$ to $S_1$, you only have to consider the invariant subspace of smooth vecto …

I will try to answer the question as far as I have understood it. Please comment. Clifford's theorem Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. The groups $G$ resp. $G/H$ act on …

I suggest the following lecture notes of Bruhat: www.math.tifr.res.in/~publ/ln/tifr14.pdf Chapter 3 & 4 should answer most of your questions. For example, there are statements like this : Proposi …