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Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t geq 2$ in any dimension $d geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t geq 6$.
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We study a correction factor for Kac-Moody root systems which arises in the theory of $p$-adic Kac-Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials $m_lambda in mathbb{Z}[t]$ indexed by positive imaginary roots $lambda$. At $t=0$ these polynomials evaluate to the root multiplicities, so we consider $m_lambda$ to be a $t$-deformation of $mathrm{mult} (lambda)$. We generalize the Peterson algorithm and the Berman-Moody formula for root multiplicities to compute $m_lambda$. As a consequence we deduce fundamental properties of $m_lambda$.
We describe how to define observables analogous to quantum fields for the semicontinuous limit recently introduced by Jones in the study of unitary representations of Thompsons groups $F$ and $T$. We find that, in terms of correlation functions of these fields, one can deduce quantities resembling the conformal data, i.e., primary fields, scaling dimensions, and the operator product expansion. Examples coming from quantum spin systems and anyon chains built on the trivalent category $mathit{SO}(3)_q$ are studied.
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
Over a non-archimedean local field of characteristic zero, we prove the multiplicity preservation for orthogonal-symplectic dual pair correspondences and unitary dual pair correspondences.
In this joint introduction to an Asterisque volume, we give a short discussion of the historical developments in the study of nonlinear covering groups, touching on their structure theory, representation theory and the theory of automorphic forms. This serves as a historical motivation and sets the scene for the papers in the volume. Our discussion is necessarily subjective and will undoubtedly leave out the contributions of many authors, to whom we apologize in earnest.
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