Some genuine small representations of a nonlinear double cover

Tsai, Wan-Yu

doi:10.1090/tran/7351

Some genuine small representations of a nonlinear double cover
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Trans. Amer. Math. Soc. 371 (2019), 5309-5340 Request permission

Abstract:

Let $G$ be the real points of a simply connected, semisimple, simply laced complex Lie group, and let $\widetilde {G}$ be the nonlinear double cover of $G$. We discuss a set of small genuine irreducible representations of $\widetilde {G}$ which can be characterized by the following properties: (a) the infinitesimal character is $\rho /2$; (b) they have maximal $\tau$-invariant; (c) they have a particular associated variety $\mathcal {O}$. When $G$ is split, we construct them explicitly. Furthermore, in many cases, there is a one-to-one correspondence between these small representations and the pairs (genuine central characters of $\widetilde {G}$, real forms of $\mathcal {O}$) via the map $\widetilde {\pi } \mapsto (\chi _{\widetilde {\pi }}, AV(\widetilde {\pi }) )$.

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