# Journal of the Ramanujan Mathematical Society

Volume 38, Issue 3, September 2023 pp. 237–263.

The twining character formula for reductive groups

**Authors**:
Jackson Hopper

**Author institution:**
Department of Mathematics, University of Maryland, College Park

**Summary: **
Let Ĝ be a connected reductive group over an algebraically closed field
with a pinning-preserving outer automorphism σ. Jantzen's twining
character formula relates the trace of the action of σ on a
highest-weight representation V{µ} of Ĝ to the character of a
corresponding highest-weight representation (V{σ})µ of a related
group {Ĝσ,∘}. This paper extends the methods of Hong's geometric
proof for the case Ĝ is adjoint, to prove that the formula holds for all
connected reductive groups, and examines the role of additional hypotheses.
In the final section, it is explained how these results can be used to draw
conclusions about quasi-split groups over a non-Archimedean local field. This
paper thus provides a more general geometric proof of the Jantzen twining
character formula and provides some apparently new results of independent
interest along the way.

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