All Questions
10 questions
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Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650
3
votes
1
answer
310
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Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
- 161
2
votes
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82
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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
1
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133
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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
2
votes
0
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156
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Spherical harmonics, $\frak{sl}_2$, and algebra gradings
Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
- 141
3
votes
1
answer
277
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Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 145
6
votes
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190
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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
3
votes
0
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130
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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
5
votes
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167
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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
1
vote
1
answer
189
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Dominant weights appear in Discrete Series
If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...
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