We investigate the group $${\mathcal {H}}_{\mathbb {C}}$$ H C of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H . Irreducible representations of the Weyl–Schrödinger type on the space $$L^2_\chi $$ L χ 2 of quadratically integrable $${\mathbb {C}}$$ C -valued functions are described. Integrability is understood with respect to the projective limit $$\chi =\varprojlim \chi _i$$ χ = lim ← χ i of probability Haar measures $$\chi _i$$ χ i defined on groups of unitary $$i\times i$$ i × i -matrices U ( i ). The measure $$\chi $$ χ is invariant under the infinite-dimensional group $$U(\infty )=\bigcup U(i)$$ U ( ∞ ) = ⋃ U ( i ) and satisfies the abstract Kolmogorov consistency conditions. The space $$L^2_\chi $$ L χ 2 is generated by Schur polynomials on Paley–Wiener maps. The Fourier-image of $$L^2_\chi $$ L χ 2 coincides with the Hardy space $${H}^2_\beta $$ H β 2 of Hilbert–Schmidt analytic functions on H generated by the correspondingly weighted Fock space $$\varGamma _\beta (H)$$ Γ β ( H ) . An application to heat equation over $${\mathcal {H}}_{\mathbb {C}}$$ H C is considered.

中文翻译：

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无限维海森堡群的外尔-薛定谔表示

我们研究了复合海森堡矩阵的 $${\mathcal {H}}_{\mathbb {C}}$$ HC 群，其条目来自无限维复数希尔伯特空间 H 。描述了二次可积 $${\mathbb {C}}$$ C 值函数在空间 $$L^2_\chi $$ L χ 2 上的 Weyl-Schrödinger 类型的不可约表示。可积性被理解为投影极限 $$\chi =\varprojlim \chi _i$$ χ = lim ← χ i 的概率 Haar 测度 $$\chi _i$$ χ i 定义在幺正 $$i\times 组上i$$ i × i - 矩阵 U ( i )。测度 $$\chi $$ χ 在无限维群 $$U(\infty )=\bigcup U(i)$$ U ( ∞ ) = ⋃ U ( i ) 下是不变的，并且满足抽象的 Kolmogorov 一致性条件. 空间$$L^2_\chi $$ L χ 2 由Paley-Wiener 映射上的Schur 多项式生成。$$L^2_\chi $$ L χ 2 的傅立叶图像与 Hilbert-Schmidt 解析函数的 Hardy 空间 $${H}^2_\beta $$ H β 2 重合，Hilbert-Schmidt 解析函数由相应加权的 Fock 生成空间 $$\varGamma _\beta (H)$$ Γ β ( H ) 。考虑了在 $${\mathcal {H}}_{\mathbb {C}}$$ HC 上的热方程的应用。