Coalescence of shtuka legs

Let $\zeta:I\to J$ be a map of finite sets. It defines a diagonal map

\Delta_\zeta:X^J\longrightarrow X^I, \qquad (x_j)_{j\in J}\longmapsto(x_{\zeta(i)})_{i\in I}.

Coalescence of shtuka legs is the canonical fusion isomorphism that identifies the pullback of the $I$ -leg shtuka cohomology along $\Delta_\zeta$ with the $J$ -leg cohomology in which all legs in one fiber of $\zeta$ have merged.

Representation-theoretic label

If $W$ is a representation of the dual group $\widehat G^I$ , restriction along

\widehat G^J\longrightarrow\widehat G^I, \qquad (g_j)\longmapsto(g_{\zeta(i)})

gives a representation $W^\zeta$ of $\widehat G^J$ . Writing $\mathcal H_{I,W}$ for the relevant cohomology sheaf, coalescence has the form

\Delta_\zeta^*\mathcal H_{I,W} \simeq \mathcal H_{J,W^\zeta}.

Geometric origin

The isomorphism comes from the factorization or fusion structure of the Beilinson–Drinfeld affine Grassmannian and the geometric Satake equivalence. It remains meaningful on diagonals where modifications collide; it is more than the evident identification on the open locus of distinct legs.

Excursion-operator role

Starting with no legs, an invariant vector $x:\mathbf 1\to W$ creates several coincident legs. Coalescence separates or groups the representation labels, partial Frobenius supplies independent Galois actions, and a covector $\xi:W\to\mathbf 1$ annihilates the legs. Functoriality of coalescence is essential for the relations among excursion operators.

References

Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” Chapter 4. arXiv.
Beilinson and Drinfeld, Quantization of Hitchin's Integrable System and Hecke Eigensheaves, factorization construction.