Generalized BGG resolutions and Blattner's formula in type A (Q6623495)

In the paper under review, the author found a unexpected correspondence between the classical invariant theory of \(GL_n(\mathbb{C})\) and \(\mathfrak{k}\)-type multiplicities in discrete series representations of \(SU(n,m)\).\N\NBy Howe duality, the polynomial functions on the space of the sum of \(q\) copies of the defining representation and \(p\) copies of its dual decompose under the joint action of \(GL_n(\mathbb{C})\) and \(\mathfrak{gl}_{p+q}\). The modules for \(\mathfrak{gl}_{p+q}\) are infinite-dimensional and their structure is complicated. When \(n< p+q\), the structure of these modules is less transparent. \textit{T. J. Enright} and \textit{J. F. Willenbring} [Ann. Math. (2) 159, No. 1, 337--375 (2004; Zbl 1087.22011)] found that these modules have finite resolutions in terms of generalized Verma modules (GVMs).\N\NIn this paper, the author showed that these resolutions can be derived from the coefficients in a formal series arising in an entirely different setting -- discrete series representations of \(SU(n,p+q)\) in the case of two noncompact simple roots. More precisely, he found that the signed multiplicities of the GVMs in the resolution coincide with the values of Blattner's formula for the \(K\)-type multiplicities in appropriately chosen discrete series representations of \(SU(n, p + q)\).