Karl-Hermann Neeb, Hadi Salmasian
The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie supergroups using the method of coadjoint orbits.
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For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathcal A:=\mathcal A(G,\mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,\mathfrak g)$ and nondegenerate $*$-representations of $\mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, …
We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of $\mathbb Z_2^n$-graded Lie supergroups.
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
For a broad class of Frechet-Lie supergroups we prove that there exists a correspondence between positive definite smooth superfunctions and matrix coefficients of unitary representations. We also give a characterization of linear functionals on the corresponding universal enveloping algebra which correspond to matrix coefficients of analytic unitary representations.
We prove that the categories of smooth and analytic unitary representations of Banach--Lie supergroups are well-behaved under restriction functors, in the sense that the restriction of a representation to an integral subsupergroup is well-defined. We also prove that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach--Lie supergroups. They extend the known results for finite dimensional Lie supergroups.…
For a locally convex Lie group with the Trotter property, we prove that the space of k-times differentiable vectors of a unitary representation is equal to the intersection of domains of k-fold products of the Lie algebra action. The result also holds for continuous representations of locally exponential groups on metrizable locally convex spaces. As an application, we extend a key stability theorem in the representation theory of Lie supergroups beyond the Banach case.
Geometric quantization and unitary highest weight Harish-Chandra supermodules
本文把几何量子化推广到超设置，对逆步李超群的实形式在伪Kähler超流形上的作用构造酉表示，得到最高权Harish-Chandra超模并建立Gelfand模型，且证明量子化与辛约化可交换。
Numerous Lie supergroups do not admit superunitary representations except the trivial one, e.g., Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of superunitary representation, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg gro…
On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators
该研究围绕格拉斯曼代数相关的涅夫 - 施瓦茨李超代数展开，通过定义其格拉斯曼化及复化，明确描述相关超群和半群，在参数不等式条件下，将超群的酉表示扩展到半群，且不要求读者有相关先验知识。
Let $(G,θ)$ be a Banach--Lie group with involutive automorphism $θ$, $\g = \fh \oplus \fq$ be the $θ$-eigenspaces in the Lie algebra $\g$ of $G$, and $H = (G^θ)_0$ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup $S \subeq G$ of the form $S = H \exp(W)$, where $W$ is an open $\Ad(H)$-invariant convex cone in $\fq$ and the polar map $H \times W \to S, (h,x) \mapsto h \exp x$ is a diffeomorphism. Any such semigroup carries an involution * satisfying $(h\exp x)^* = (\exp x) h^{-1}$. Our central result, generalizing the Lüscher--Mack Theorem for finite …
Quasifinite representations of classical Lie subalgebras of W_{1+infty}
我们证明了在保主阶的圆上的微分算子代数的反对合sigma_{\pm}有两个共轭对合。对由-sigma_{\pm}固定的李子代数的中心扩张\hat{D}^{\pm}的不可约拟有限最高权表示进行了分类，并找到了酉表示.我们用截断多项式代数C [u]/（u ^{m +1}）及其B、C、D型经典李子代数上具有n个非零对角线的无限矩阵李代数的中心扩张的最高权表示来实现它们.得到了\hat{D}^{\pm}的正本原表示（包括所有酉表示）的特征标公式.我们还实现了\hat{D}^{\pm}的一类自由域的本原表示，并建立了这些本原表示与有限维李群和超群的有限维不可约表示之间的对偶结果.证明了\hat{D}^+的真空模V_c具有顶点代数结构，并建立了半整中心荷c的V_c与W-代数之间的关系.
Regular Functions on the K-Nilpotent Cone
本文推导了复约化代数群实形式相关的K-幂零锥不变子簇的正则函数环作为表示的公式，为计算Hodge滤过和猜想幺正对偶描述提供了关键要素。
We give a general description of the moment cone associated with an arbitrary finite-dimensional unitary representation of a compact, connected Lie group in terms of finitely many linear inequalities. Our method is based on combining differential-geometric arguments with a variant of Ressayre's notion of a dominant pair. As applications, we obtain generalizations of Horn's inequalities to arbitrary representations, new inequalities for the one-body quantum marginal problem in physics, which concerns the asymptotic support of the Kronecker coefficients of the symmetric group, and a geometric in…
Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement generalized the well-known theorems of Coburn and Douglas. In this note we prove the reverse. If all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic, then S is a positive cone of G. Also we consider G = Z\times Z and prove that if S induces total order on G, then there exist at least two unitarily not …
A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semi-bounded if the corresponding operators $i\ddπ(x)$ from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting conne…
Wave Front Sets of Nilpotent Lie Group Representations
本文研究幂零连通单连通李群的酉表示，证明其波前集等于该表示轨道支撑的渐近锥，建立了表示论中波前集与Kirillov对应下余伴随轨道的几何联系。
Partial convex hulls of coadjoint orbits and degrees of invariants
本文通过引入部分凸包概念定义紧李群余伴随轨道的两个数值不变量，证明其与 Littlewood-Richardson 锥相关，并给出不可约酉表示上不变多项式次数的下界。
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