Semisimple representatives in a torus normalizer

Let $H$ be a complex linear algebraic group whose identity component $H^\circ$ is reductive. A semisimple element $s\in H$ normalizes some maximal torus of $H^\circ$ . Since maximal tori of $H^\circ$ are $H^\circ$ -conjugate, after conjugation one may place $s$ in the normalizer

N_H(T)

of a fixed maximal torus $T\subset H^\circ$ .

Based data

A representative in $N_H(T)$ induces an automorphism of the root system. Composing with a Weyl-group element can arrange preservation of a chosen positive system when the induced component admits such a representative. This is a statement about a component and its twisted conjugacy, not the ordinary assertion that every semisimple element of a connected group lies in $T$ .

Connected special case

If $H$ is connected reductive, every semisimple element is conjugate into $T$ itself. Its ordinary conjugacy class is then a Weyl orbit in $T$ .

Langlands role

This is the normalizer-representative principle used in the letter for a disconnected group such as $\widehat G$ $\rtimes\Gamma$ . In an $L$ -group, an unramified parameter lies in a Frobenius coset $\widehat G\rtimes\operatorname{Frob}$ . Conjugacy inside that coset is a twisted-conjugacy problem. A torus-normalizer representative makes the root-theoretic Satake description possible, but modern formulations use the invariant quotient of the Frobenius fiber rather than treating the coset as a connected reductive group.

Attribution warning

The source letter invokes Borel–Mostow in a specific disconnected algebraic-group setting. Variants in the literature have different hypotheses on the component group and on semisimplicity. The connected special case above is unconditional; any stronger “dominant representative” assertion should be cited with its exact version.

References

A. Borel and G. D. Mostow, “On semi-simple automorphisms of Lie algebras,” Annals of Mathematics 61 (1955), 389–405. DOI.
T. A. Springer, Linear Algebraic Groups, second edition, Birkhäuser, 1998, §§7–8.