Search Results

Yes, your desired equality is true: regarding the left regular representation as $$ \operatorname{Ind}_{\{0\}}^{\mathbf R}1, $$ it becomes a special case of the characterization of smooth vectors in i …

I am not an expert, but it seems that Geck's constructions specialize the Lusztig-Kashiwara "canonical" or "crystal" bases of highest weight modules. These also have a Littelmann "path model", related …

As in Chriss-Ginzburg p. 130, pick a regular semisimple $h\in\mathfrak h$ (Cartan subalgebra) and write $\mathfrak g_a$ for the eigenspace of $\text{ad}_h$ belonging to eigenvalue $a$. Invariance of t …

I don't believe that weak continuity is "the only place in which $\pi$ being a homomorphism plays a role". In fact, before you can even talk about weak continuity of the resulting representation, you …

Quoth Dixmier, Enveloping algebras, p. xii: "But a deeper study reveals the existence of an enormous number of irreducible representations of [the 3-dimensional Heisenberg algebra]. It seems that thes …

This is true and due to Béla von Szőkefalvi-Nagy, Über meßbare Darstellungen Liescher Gruppen (1936). Generalized to finite-dimensional representations of locally compact groups in A. Weil, L'intégrat …

For complementary series, I'd recommend §V.4 of Sugiura (available on the internet) for a very careful description of their embedding in nonunitary principal series. In particular, Prop. 4.6 explains …

Georg Frobenius, Über die Primfactoren der Gruppendeterminante, Sitzungsber. Akad. Berlin (1896) 1343-1382. The theorem is announced at the beginning, p. 1344: Der Grad $f$ ist ein Divisor der Ord …

In your inequality you need to evaluate the $\pi$'s somewhere! Unless I am mistaken unpacking Fell (1962), Theorem 2.2 and Remark following, $[\pi_j]\to[\pi]$ means that for every choice of an $\vare …

I would suggest Procesi's Lie Groups, as a text that introduces algebraic groups with minimal prerequisites. Chapter 7 "Algebraic Groups" is a quick introduction to algebraic groups. In this chapt …

(Addressing only the title question.) There is a short proof avoiding Peter-Weyl and the theory of compact operators. It is due to Nachbin and is reproduced in Hewitt-Ross, Abstract Harmonic Analysis …

The unitary dual of SO(1,4) was computed by Dixmier (1961); that of SO(2,3), or the locally isomorphic Sp(2,R), by Angelopoulos (1981) and Nzoukoudi (1983).

Such a $\Phi$ is not going to exist — nor is $H$ going to have nonzero $\mu(H)$ — unless $H$ is open as well as closed in $G$. This is the very case treated in Mackey (1951, Part II): then $G/H$ is di …

Knapp (1986, p. 736) attributes the Plancherel formula for real-rank-one groups to Okamoto (1965), Hirai (1966), and Harish-Chandra (1966). More details in Sally-Warner (1973), Miatello (1979).

I may be completely off-base, and would be happy to be proved wrong, but I believe you are veering close to problems that are reputedly intractable. Namely, to simplify your simplest example even furt …