Abstract We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups $$K = mathrm {O}_N,$$ K = O N , $$mathrm {SO}_N,$$ SO N , and $$mathrm {USp}_N,$$ USp N . To that end, we start from the Clifford–Weyl algebra in its canonical realization on the complex $$ancyscript {A}_V$$ A V of holomorphic differential forms for a $$mathbb {C} $$ C -vector space $$V_0,$$ V 0 . From it we construct the Fock representation of an orthosymplectic Lie superalgebra $$mathfrak {osp}$$ osp associated to $$V_0,$$ V 0 . Particular attention is paid to defining Howe’s oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of $$mathfrak {sp} subset mathfrak {osp}$$ sp ? osp . In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale–Weil–Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let $$V_0 = mathbb {C}^n otimes mathbb {C}^N$$ V 0 = C n ? C N where $$mathbb {C}^N$$ C N is equipped with the standard K -representation, and focus on the subspace $$ancyscript {A}_V^K$$ A V K of K -equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer $$mathfrak {g}$$ g of $$mathrm {Lie}(K)$$ Lie ( K ) in $$mathfrak {osp}$$ osp . We identify the K -Haar expectation of n ratios with the character of this $$mathfrak {g} $$ g -representation, which we show to be uniquely determined by analyticity, Weyl-group invariance, certain weight constraints, and a system of differential equations coming from the Laplace-Casimir invariants of $$mathfrak {g},$$ g . We find an explicit solution to the problem posed by all these conditions. In this way, we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers $$N ge 1$$ N ≥ 1 . This completes earlier work of Conrey, Farmer, and Zirnbauer for the case of $$mathrm {U}_N,$$ U N .
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机译：摘要我们计算经典李群$$ K = mathrm {O} _N ，$$ K = ON，$$ mathrm {SO} _N ，$$ SO N的随机特征多项式比率的Haar集合平均。和$$ mathrm {USp} _N ，$$ USp N。为此，我们从Clifford-Weyl代数开始，它是关于$$ mathbb {C} $$ C-矢量空间$的全纯微分形式的复数$$ fancyscript {A} _V $$ AV的规范实现的。 $ V_0 ，$$ V 0。由此，我们构造了与$$ V_0 ，$$ V 0相关联的正统李超代数$$ mathfrak {osp} $$ osp的Fock表示。特别要注意定义Howe的振子半群，以及表示部分对$$ mathfrak {sp} subset mathfrak {osp} $$ sp？的李代数表示进行幂运算的表示。 osp。在此过程中，通过将半群表示推到其边界并通过连续性进行争论，我们提供了元偏元群的Shale-Weil-Segal表示的构造。为了处理n个特征多项式比率的乘积，我们让$$ V_0 = mathbb {C} ^ n otimes mathbb {C} ^ N $$ V 0 = C n？ C N，其中$$ mathbb {C} ^ N $$ C N配备了标准K-表示，并关注子空间$$ fancyscript {A} _V ^ K $$ A V K为K-等价形式。通过Howe对偶，这是扶正器的最高权重不可约表示。$$ mathfrak {g} $$ g $$ mathrm {Lie}（K）$$ Lie（K）以$$ mathfrak {osp} $$ osp。我们用这个$$ mathfrak {g} $$ g-表示的特征来标识n个比率的K -Haar期望，我们证明这是由解析性，Weyl群不变性，某些权重约束和一个系统唯一确定的来自$$ mathfrak {g} ，$$ g的Laplace-Casimir不变量的微分方程组。我们找到了解决所有这些情况造成的问题的明确解决方案。这样，我们证明对于所有整数$$ N ge 1 $$ N≥1，上述Haar期望均由Weyl型字符公式表示。这样就完成了$$ / mathrm {U} _N ，$$ U N的情况下Conrey，Farmer和Zirnbauer的早期工作。
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