All Questions
15 questions with no upvoted or accepted answers
7
votes
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answers
420
views
What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?
Here is the story as I see it.
Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
- 557
7
votes
0
answers
502
views
Relation between Lie group characters and spherical functions on symmetric spaces
Setup: Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real ...
- 32.5k
6
votes
0
answers
190
views
Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$
Lie algebraically, the eigenvalue of the spherical function
\begin{align*}
\phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*)
\end{align*}
...
- 83
6
votes
0
answers
184
views
Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
- 25.8k
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
5
votes
0
answers
167
views
Plancherel formula for $L^2(G/N)$
Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...
- 51
4
votes
0
answers
230
views
How to decompose the left regular representation of a real reductive group?
In Dixmier ($C^*$-algebras), the Plancherel theorem states (I will not mention the right regular representation even though the theorem does talk about it):
Let $G$ be unimodular, $\lambda$ be the ...
- 561
4
votes
0
answers
173
views
Ring of SO(n)-invariant differential operators on M_n,m
I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...
- 41
3
votes
0
answers
75
views
Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
- 81
3
votes
0
answers
78
views
Decay of Fourier coefficients for Hölder functions on compact Lie groups
If $f$ is a complex-valued function on a compact Lie group $G$, we have a decomposition $f = \sum_\mu f_\mu$ corresponding to the Peter-Weyl decomposition $L^2(G) = \oplus_\mu (\dim \mu) V_\mu$.
For $...
- 31
3
votes
0
answers
218
views
Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?
The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ ...
- 650
3
votes
0
answers
130
views
About the purpose of introducing '"groups of Heisenberg type"
I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
- 650
2
votes
0
answers
82
views
Question on a remark in Speh's paper
I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
- 1,121
2
votes
0
answers
81
views
Fourier transform in the complex motion group
I am looking for a reference that deals with the unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$ i.e., the semi-direct product of $\mathbb C^2$ with the special unitary group $K=...
- 650
1
vote
0
answers
133
views
Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
- 1,121