On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups. (English) Zbl 1378.11061

Let \(F\) denote a non-Archimedean local field of characteristic zero and let \(G\) be a connected reductive group over \(F\). Also, let \(Z\) be a maximal \(F\)-split torus of the center of \(G\).
The formal degree \(d_{\pi}\) of an irreducible unitary square-integrable representation \(\pi\) of \(G\) on a Hilbert space \(V\) is the Haar measure on \(Z\mathbb G\) such that \[ \int_{Z\mathbb G}(\pi(g)v_1, v'_1)(\pi(g)v_2, v'_2) d_{\pi}g = (v_1, v'_2)(v_2, v'_1) \] holds for any \(v_1, v_2, v'_1, v'_2 \in V\) (here \((\cdot, \cdot)\) stands for a \(G\)-invariant inner product on \(V\)).
In the paper under the review, authors prove an explicit formula for the formal degree for the metaplectic group \(Mp_n\), a double cover of the rank \(n\) symplectic group. Such a formula has been conjectured in [K. Hiraga et al., J. Am. Math. Soc. 21, No. 1, 283–304 (2008; Zbl 1131.22014)], and extends Harish-Chandra’s formula to the non-Archimedean case and also fits well with Langlands’s conjecture on Plancherel measures.
Authors use the local integrals and descent method to reduce the formal degree conjecture for generic discrete series (i.e., irreducible square-integrable representations) of \(Mp_n\) to a known local identity, obtained in [the last two authors, Algebra Number Theory 11, No. 3, 713–765 (2017; Zbl 1418.11076)]. Also, the theta correspondence methods enable authors to prove the formal degree conjecture for generic discrete series of the split odd special orthogonal group \(\mathrm{SO}_{2n+1}\). Then Arthur’s work on the local Langlands correspondence implies the formal degree conjecture for general discrete series representations.


11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E35 Analysis on \(p\)-adic Lie groups
11F55 Other groups and their modular and automorphic forms (several variables)
11F27 Theta series; Weil representation; theta correspondences
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