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Summary

We prove the existence of regular semi-simple elements, then of regular unipotent elements. We give properties of Gelfand–Graev representations and their duals. We compute their values on regular unipotent elements using Gauss sums. We decompose Gelfand–Graev representations into sums of regular characters and use them to prove the disjunction of Deligne–Lusztig characters corresponding to distinct rational semi-simple classes. The chapter ends with the character table of SL2.

For an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$. For $ \alpha \in \mathcal {I}(X)$, let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$. For an arbitrary $ \alpha \in \mathcal {I}(X)$, we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$, describe the regular elements of $C(\alpha )$, and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$, we determine the structure of $C(\alpha )$in terms of Green’s relations.

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12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters

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CENTRALISERS IN THE INFINITE SYMMETRIC INVERSE SEMIGROUP

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12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters

Summary

CENTRALISERS IN THE INFINITE SYMMETRIC INVERSE SEMIGROUP