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Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space …

I just wrote this answer on your last question. To summarize, you can't have an irreducible representation of a compact group that is infinite dimensional, unless the representation space is very exot …

For a general real reductive group, all irreducible admissible $({\mathfrak g},K)$-modules are quotients of parabolically-induced discrete series (or limits thereof) representations (where we allow " …

For a reductive Lie group, the character characterizes an irreducible admissible representation up to infinitesimal equivalence. Referring to Knapp's "Representation Theory, etc", Proposition 10.5 say …

There's a slightly subtle point near here of which some people are not aware: that it is dangerous (perhaps even nonsensical) to compare algebraic numbers under various different completions. So, to t …

This is kind of a complicated question, since there isn't really a single good answer. We begin with a simple Lie group $G$ (for simplicity!). On the one hand, we hopefully have a description of the …

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, …

This is precisely the content of Harish-Chandra's theorem ("Automorphic forms on Semisimple Lie Groups", LNM 68, 1968), proven for general reductive groups: Fix a finite-dimensional representation $\ …

The standard way to decompose $L^2(G)$ (where $G$ can be a wide variety of things) is to decompose a dense subspace (usually Schwartz or compactly supported), then prove the Plancherel formula, which …

I'm trying to use the language of strata to organize $K$-types of irreducible smooth representations of $GL(2)$ (and then hopefully prove things). Unfortunately, I'm still new to it, so I might be mak …

In this paper, Shintani proves the Casselman-Shalika(-Shintani) formula for GL(n). This preceded Casselman-Shalika's paper by a few years. Several of Cogdell's expository articles on L-functions have …

The answer depends on both the nature of the subgroup and the nature of the representation. For example, with $G=SL_2(\mathbb R)$, a discrete series discretely decomposes when restricted to $H=SO(2)$, …

I think this can be seen directly. Let's work over $G=PGL(n)$, so I don't have to keep repeating "modulo the center". Recall that supercuspidal representations can be realized as subrepresentations in …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\res{res}$For an irreducible smooth (generic) representation $\pi$ of $G=\GL_2(k)$ with central character $\omega$, where $k$ is a $p$-adic field, we de …

Let $W$ be the normalized spherical Whittaker function attached to a spherical representation $\pi$ on $GL_n(k)$, where $k$ is a $p$-adic field and $n\ge 3$. I'm faced with the slightly-odd integral …