# Questions tagged [unitary-representations]

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### Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. A function $\lambda\colon G\to \mathbb C$ is an ...
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### Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...
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### Kazhdan Property T of semisimple Lie groups

I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN). I want to ...
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### Relation between topological and differentiable Lie group cohomology for unitary modules

It is well known, that if $V$ is a $C^\infty$ $G$-module, then the differentiable cohomology groups $H^*_{d}(G, V)$ are isomorphic to $H^*(\mathfrak{g}, K, V)$. I am interested if the result can be ...
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### Weyl theorem for non specified primitive root of unity

Let $\omega=e^{2i \pi/p}$. Weyl theorems give all representations of matrix algebra span by $A,B$ such that either $AB=\omega BA, A^p=B^p=I$, or $(k,l)\mapsto A^kB^l$ is a irreducible ...
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### Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity

Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$. Define the $p$ fourrier transform ...
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### Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...
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### tensor product of massless Poincare representations

Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles? Massless ...
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### Restriction that contains a trivial representation

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$ such that $(G,G')$ is a symmetric pair. If $\pi$ is an infinitely dimensional unitary representation of $G$, ...
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### Tensor products of unitary irreducible representations of $SU(2,2)$

What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...
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### Does the supercuspidal representation becomes tempered?

I am really wondering whether supercuspidal representation may become tempered representation. If it is not true for all classical group, is it especially true for unitary group? If it is not true ...
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### Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
### 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 ...