# Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?

Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence still raises questions for me.

Without introducing too much notation (which varies a lot in different sources), I'll recall the key result of the early paper here by Hotta and Springer on what they call "specialization". They work over a suitable algebraically closed field with the cohomology of certain subvarieties of a flag variety $\mathcal{B}$ of a (say) simple algebraic group $G$ with Lie algebra $\mathfrak{g}$. If $u \in G$ is a unipotent element, write $\mathcal{B}_u$ for the fixed point set of $u$: this is essentially the set of Borel subgroups containing $u$, which as a variety is determined up to isomorphism by the conjugacy class of $u$. (Some versions instead use nilpotent elements of $\mathfrak{g}$, which is equivalent to this set-up if the characteristic is good.)

Now the obvious inclusion $\mathcal{B}_u \hookrightarrow \mathcal{B}$ induces a reverse map $\varphi$ on cohomology (which can be classical or etale). Here the cohomology typically vanishes in odd degrees, while the top degree $2d(u)$ is equal to twice the dimension of $\mathcal{B}_u$. This degree (sometimes abbreviated $d(u)$) is the main concern of the Springer correspondence, which realizes certain irreducible "Springer characters" of the Weyl group $W$ in a unique top cohomology group (where $W$ acts naturally even though it doesn't usually act on $\mathcal{B}_u$). In case $u$ has a disconnected centralizer in $G$, the characters of the finite group of connected components complicate the Springer correspondence further; but a Springer character always occurs as the unique irreducible summand of the top cohomology belonging to the trivial character of the component group.

Given this set-up, the main theorem of Hotta-Springer says that $\varphi$ maps $H^{2d(u)}(\mathcal{B})$ onto the fixed points of the component group of $u$ in $H^{2d(u)} (\mathcal{B_u})$. However, there are exceptions to this surjectivity in some lower degrees, which seem to be related indirectly to Lusztig's "generalized Springer correspondence".

When $u$ is a special unipotent (in Lusztig's sense), are there examples of such failures of surjectivity?

Here the definition of "special" is unfortunately roundabout, related to Lusztig-Spaltenstein duality and the notion of "unipotent pieces". But when $G$ is a special linear group, all unipotents are special. In that case I'm just asking whether the Hotta-Springer theorem has a stronger statement.

[ADDED] Daniel Juteau reminds me that the case of general and special linear groups is discussed more directly in $\S2$ of the Hotta-Springer paper, where surjectivity of their graded $W$-equivariant specialization map on cohomology is proved in (2.3). For these groups one has the advantage that all centralizers of unipotent elements are connected (so the component groups are all trivial), while the unipotent classes themselves are all Richardson (hence special) and of standard Levi type. Here the Weyl group is a symmetric group, and all of its irreducible characters are Springer characters.

It would be especially interesting to sort out (using for example the old Beynon-Spaltenstein tables) exactly what happens in the extreme type $E_8$: here $G$ has 70 unipotent classes, with 46 being special, and one special class has component group $S_5$ (comprising a special piece along with six of the non-special classes in its closure).