Vogan duality for nonlinear type \(B\) (Q2889326)

Let \(\mathbb G=\mathrm{Spin}(4n+1)\) be the connected, simply connected complex (reductive) Lie group of type \(B_{2n}\), and let \(G=\mathrm{Spin}(p,q)\) with \(p+q=4n+1\) denote a real form of \(\mathbb G\). If \(q\notin\{0,1\}\) then \(G\) has a nonlinear two-fold covering group \(\widetilde G=\widetilde{\mathrm{Spin}}(p,q)\). The main result of this paper is a certain special symmetry in the set of genuine representation parameters for the various \(\widetilde G\) at half-integral (non-singular) infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized Hecke modules and ultimately results in a character multiplicity duality for the genuine characters of \(\widetilde G\).NEWLINENEWLINEGiven a (genuine) irreducible admissible representation \(\pi\) of \(\widetilde G\), let \(X(\pi)\) be the standard module that contains \(\pi\) as its Langlands subquotient. Because the character of a standard module is more readily computable than that of an irreducible, it is of interest to study the composition factors of \(X(\pi)\) via expressions of the form NEWLINE\[NEWLINE X(\pi)=\sum_{\sigma}m(\sigma,\pi)\sigma.NEWLINE\]NEWLINE Here the \(\sigma\) run over the \textit{block} of representations containing \(\pi\): a certain finite set of irreducible representations of \(\widetilde G\) with the same infinitesimal character as that of \(\pi\). The \(m(\sigma,\pi)\) are nonnegative integers. This sum may be interpreted as an identity in an appropriate Grothendieck group, or as a character identity. Similarly, we have the problem of finding an expression for \(\pi\) in terms of the standard representations (now with possibly negative coefficients) NEWLINE\[NEWLINE\pi=\sum_{\sigma}M(\sigma,\pi)X(\sigma).NEWLINE\]NEWLINE Given a block \(\mathcal B=\{\pi_1,\pi_2,\dots,\pi_r\}\) of representations of \(\widetilde G\) (or their parameters), the author defines a group \(\widetilde G'\) and a block \(\mathcal B^{\vee}=\{\sigma_1,\dots,\sigma_r\}\) for \(\widetilde G'\). In addition to having the same infinitesimal character, the elements of a block also have the same central character. Like \(\widetilde G\), the group \(\widetilde G'\) is the non-algebraic two-fold cover of a real form of \(\mathbb G\), and depends on the infinitesimal character and central character of \(\mathcal B\). The block \(\mathcal B^{\vee}\) then has the same infinitesimal character and the opposite (genuine) central character. This construction ultimately gives a bijection (\(\pi_i\mapsto\sigma_i\)) between all genuine representation parameters of all real forms at the given infinitesimal character. The parameters in a block form a basis of a certain module for an extended Hecke algebra, generated by cross actions and Cayley transforms, and the dual block supports a dual Hecke module. The combinatorics of these Hecke modules give rise to an algorithm (generalized Kazhdan-Lusztig algorithm) for computing the multiplicities \(M(\pi_i,\pi_j)\) and \(m(\pi_i,\pi_j)\). The main theorem asserts that the bijection results in a character multiplicity duality: for each block \(\mathcal B\) and each pair \((i,j)\) of indices, we have NEWLINE\[NEWLINE M(\sigma_i,\sigma_j)=\pm m(\pi_j,\pi_i),NEWLINE\]NEWLINE with the sign explicitly computable.NEWLINENEWLINEFor linear real reductive Lie groups, the corresponding result is due to \textit{D. A. Vogan} [Duke Math. J. 49, 943--1073 (1982; Zbl 0536.22022)]; in this case, the dual block consists of representations of a certain real form of the Langlands dual group. The paper under review represents a part of ongoing work on nonlinear double covers by a number of authors, such as the paper by \textit{D. A. Renard} and \textit{P. E. Trapa} [Represent. Theory 4, 245--295 (2000; Zbl 0962.22008)] who introduced the idea of generalized Hecke algebras to extend the Kazhdan-Lusztig algorithm to the case of such groups, and formulated a duality theorem for the metaplectic group; and by \textit{J. Adams} and \textit{P. E. Trapa} [Compos. Math. 148, No. 3, 931--965 (2012; Zbl 1276.22004)] who have established a duality theory for nonlinear double covers of groups whose root system is simply laced.