Search Results

Various special constructions make the point for particular groups, which tends to convince one about the general case. E.g., for $G=SL(2,\mathbb Q)$, a well-known (Shalika, Jacquet, c. 1970) idea is …

Maybe! My nominee is Jacquet-PS-Shalika's "essential vector", that is, the (unique up to scalars) vector fixed by the compact subgroup of $GL(n,k_v)$ consisting of matrices $\pmatrix{A&b\cr c&d}$ with …

This left-right business is largely an accident of notation/language/writing, which makes it all the more difficult to distinguish genuine mathematical features from accidents of writing left-to-right …

Depending what one means by "intuition", the computation of the behavior of the center $\mathfrak z$ of $U\mathfrak g$ on unramified principal series repns not only suggests the form of the Harish-Cha …

As @user19918273 noted, uniqueness fails immediately: somewhat more generally, for $SL_n(k_v)$ for non-archimedean $k_v$, there are $n$ conjugacy classes of maximal compacts. However, there is a uniqu …

For spherical principal series $I_s$ (non-normalized) induced from $\pmatrix{a & * \cr 0 & a^{-1}} \rightarrow |a|^{2s}$, the dual is $I_{1-s}$, with pairing given by integration over $K$. This is iso …

Probably the relevant work is that of Casselman and Wallach (independently) on (in effect) adjoints (right? left?) to the forgetful functor taking Lie group repns to Lie algebra repns. The keyword is …

Just for contrast/complement to the other good (upvoted) answers: in various situations I've found it sufficient, and possible, to just ask about multiplicity-free-ness of an induced repn for a restri …

In addition to other answers, especially if one is interested in (infinite-dimensional) representation theory of Lie groups, a common technical issue is that the maximal compact subgroup should meet a …

Adding some further detail to Will Sawin's answer, in some examples that are clearer than the general case: For semi-simple real Lie groups $G$ obtained by "restricting scalars" of ("classical") comp …

Although perhaps the question was directed more at finite groups: for reductive or semi-simple real Lie groups, (serious) results of Harish-Chandra (starting in the early 1950s) show that among regula …

If I understand the intent of the question: partly iterating what @Alexander Chervov's answer notes, ... For example, a theorem of Harish-Chandra asserts that the collection of irreducible admissible …

It seems unlikely, given that already the assertion is not literally true for the tensor product of two copies of the standard representation $V$ of $G=SL_2(\mathbb R)$. Namely, with $v$ a highest-wei …

In addition to the many other interesting and useful answers, and as evidence for the fruitfulness of the question (!), I do think there are a few other (maybe-interesting and maybe-useful) points to …

I can't help but add: consider the chain of $G$-equivariant natural maps, using the identification of the Lie algebra $g$ with its dual via the non-degenerate $G$-equivariant Killing form: $$ End(g) \ …