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Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \mapsto \langle \pi(g)v, w \rangle$) are in $L^2(G)$.

The formal degree $\mathrm{deg}(\pi)$ of a discrete series is defined as the "norm" of the operator "matrix coefficient", i.e. as the constant $\mathrm{deg}(\pi)$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = \mathrm{deg}(\pi)^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, $C^\star$-Algebras, Prop. 18.8.5) that for discrete series the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\lambda_\pi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \mapsto \langle \pi(g)v, w \rangle$) are in $L^2(G)$.

The formal degree $\mathrm{deg}(\pi)$ of a discrete series is defined as the "norm" of the operator "matrix coefficient", i.e. as the constant $\mathrm{deg}(\pi)$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = \mathrm{deg}(\pi)^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, $C^\star$-Algebras, Prop. 18.8.5) that for discrete series the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\lambda_\pi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have the equality $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients ($\xi^\pi_{v,w} : g \mapsto \langle \pi(g)v, w \rangle$) are in $L^2(G)$.

The formal degree $\mathrm{deg}(\pi)$ of a discrete series is defined as the "norm" of the operator "matrix coefficient", i.e. as the constant $\mathrm{deg}(\pi)$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = \mathrm{deg}(\pi)^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, $C^\star$-Algebras, Prop. 18.8.5) that for discrete series the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\lambda_\pi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \mapsto \langle \pi(g)v, w \rangle$) are in $L^2(G)$.

The formal degree $\mathrm{deg}(\pi)$ of a discrete series is defined as the "norm" of the operator "matrix coefficient", i.e. as the constant $\mathrm{deg}(\pi)$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = \mathrm{deg}(\pi)^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, $C^\star$-Algebras, Prop. 18.8.5) that for discrete series the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\lambda_\pi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is a discrete series if its matrix coefficients are in $L^2(G)$.

The, it is possible to define its formal degree $\mathrm{deg}(\pi)$ as the "norm" of the operator "matrix coefficient", i.e. as the constant $d_\pi$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = d_\pi^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, C* Algebras, Prop. 18.8.5) that in this case the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\xi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients ($\xi^\pi_{v,w} : g \mapsto \langle \pi(g)v, w \rangle$) are in $L^2(G)$.

The formal degree $\mathrm{deg}(\pi)$ of a discrete series is defined as the "norm" of the operator "matrix coefficient", i.e. as the constant $\mathrm{deg}(\pi)$ such that for all $v, w \in V_\pi$ we have $$\| \xi^\pi_{v, w} \|_2^2 := \int_{G} |\langle \pi(g)v, w \rangle|^2 dg = \mathrm{deg}(\pi)^{-1} \|v\|^2\|w\|^2.$$

It is known (e.g. Dixmier, $C^\star$-Algebras, Prop. 18.8.5) that for discrete series the formal degree matches the Plancherel measure, i.e. $$\mathrm{deg}(\pi) = \mu^{\rm Pl} (\pi).$$

I would like to relate this to the dimension of the "cohomological class" of $\pi$. As in this question, a discrete series $\pi$ is $\lambda_\pi$-cohomological for a certain $\lambda_\pi$ (its infinitesimal character if I understand correctly). This $\lambda_\pi$ is an irreducible finite-dimensional representation of $G (\mathbb C)$. Do we have $$\mathrm{deg}(\pi) = \mathrm{dim} (\lambda_\pi) \quad ?$$

References for these matters are welcome.