Hecke modification - Knowlpedia

Definition

Let $X$ be a smooth curve, $x\in X$ , and $E,E'$ principal $G$ -bundles. A Hecke modification of $E$ to $E'$ at $x$ is an isomorphism

\beta:E|_{X\setminus\{x\}}\xrightarrow{\sim} E'|_{X\setminus\{x\}}.

Thus the bundles may differ at $x$ but agree on its complement.

Relative position

After choosing a local coordinate and trivializations near $x$ , the modification gives a point of the affine Grassmannian. Its $G\lbrack\!\lbrack t\rbrack\!\rbrack$ -orbit, indexed by a dominant coweight $\lambda$ , is the relative position of the modification. The bound “relative position at most $\lambda$ ” uses the associated affine Schubert variety.

Example for GL_n

For $G=GL_n$ , a modification is a pair of vector bundles identified away from $x$ . An elementary modification can replace $E$ by the kernel of a surjection $E\to i_{x*}Q$ , changing the degree by $-\dim Q$ .

References

Vladimir Drinfeld, “Two-dimensional $\ell$ -adic representations of the fundamental group of a curve over a finite field and automorphic forms on $GL(2)$ ,” American Journal of Mathematics 105 (1983), 85–114. DOI.