Theta correspondence and Springer correspondence

In this paper, we obtain an explicit formula for the theta correspondence of unipotent principal-series representations between an even orthogonal and a symplectic group or between general linear groups over a finite field. The formula is in terms of the Springer correspondence. Along the way we prove general results about module categories of Hecke categories arising from spherical varieties, and give a similar formula for the multiplicities of the unipotent principal series representations in the function space of the spherical variety in terms of relative Springer theory.

💡 Research Summary

This paper presents a significant advancement in the field of representation theory by establishing a profound connection between two fundamental pillars: the Theta correspondence and the Springer correspondence. The primary objective of the research is to derive an explicit, computable formula for the theta correspondence of unipotent principal-series representations, moving beyond mere existence proofs to concrete algebraic formulations.

The authors focus on the correspondence between even orthogonal and symplectic groups, as well as between general linear groups over finite fields. The breakthrough lies in the utilization of the Springer correspondence—a geometric tool relating nilpotent orbits to Weyl group representations—to express the theta correspondence in a clear, explicit manner. By translating the complex algebraic interactions of dual reductive pairs into the language of the Springer correspondence, the authors provide a powerful computational framework that simplifies previously intractable problems in the study of unarmored representations.

Beyond the specific derivation of the formula, the paper extends its scope to the broader structural properties of Hecke categories. The authors investigate module categories of Hecke categories that arise from spherical varieties, which are essential generalizations of symmetric spaces. A key achievement in this section is the proof of general results regarding the multiplicities of unipotent principal-series representations within the function space of these spherical varieties.

To achieve this, the paper introduces and employs “relative Springer theory.” This theoretical framework allows the authors to express the multiplicities of representations—a purely algebraic and often difficult-to-calculate quantity—in terms of the geometric properties of relative orbits. This integration of algebraic representation theory and geometric representation theory demonstrates that the structural complexities of group representations are deeply encoded within the underlying geometric landscape of the associated varieties.

In summary, this work provides a unified and highly practical approach to understanding the relationship between different group representations. By bridging the gap between the Howe correspondence (theta correspondence) and the geometric mechanics of the Springer correspondence, the authors have equipped the mathematical community with a robust toolset for analyzing the multiplicities and correspondences of unipotent representations, paving the way for further explorations into the Hecke categories of spherical varieties.