Lusztig, George; Smelt, J. M. Fixed point varieties on the space of lattices. (English) Zbl 0779.14004 Bull. Lond. Math. Soc. 23, No. 3, 213-218 (1991). Let \(k\) be an algebraically closed field and let \(K=k((\varepsilon))\), \({\mathfrak O}=k[[\varepsilon]]\), where \(\varepsilon\) is an indeterminate. Let \(V\) be a \(K\)-vector space of dimension \(n \geq 1\) with a given volume form \(\omega\). – Let \(X\) be the set of all \({\mathfrak O}\)-lattices \({\mathcal L}\) in \(V\) such that for some \({\mathfrak O}\)-basis \(f_ 1,\ldots,f_ n\) of \({\mathcal L}\), we have \(f_ 1\wedge\ldots\wedge f_ n=\omega\). – Let \(\tilde X\) be the set of sequences of lattices \({\mathcal L}_ 0\), \({\mathcal L}_ 1,\ldots,{\mathcal L_ n}\) in \(V\) such that \({\mathcal L}_ i\) is contained with \(k\)-codimension 1 in \({\mathcal L}_{i-1}\) for \(i=1,\ldots,n\), and \({\mathcal L}_ n =\varepsilon{\mathcal L}_ 0\). (This is an affine flag manifold.) Let \(\rho:\tilde X \to X\) be the map \(\rho({\mathcal L}_ 0,\ldots,{\mathcal L}_ n)={\mathcal L}_ 0\).Let \(T\) be an endomorphism of \(V\). Let \(X^ T\) be the set of all \({\mathcal L} \in X\) such that \(T({\mathcal L}) \subset{\mathcal L}\); let \(\tilde X^ T\) be the set of all \(({\mathcal L}_ 0,\ldots,{\mathcal L}_ n) \in \tilde X\) such that \(T({\mathcal L}_ i) \subset{\mathcal L}_ i\) for all \(i\). We say that \(T\) is nil-elliptic if \(\lim_{m \to \infty}T^ m=0\) and the characteristic polynomial of \(T\) is irreducible over \(K\). – Let \(e_ 1,\ldots,e_ n\) be a basis of \(V\) such that \(e_ 1\wedge\cdots\wedge e_ n=\omega\). Let \(N:V \to V\) be the \(K\)-linear transformation given by \(N(e_ i)=e_{i- 1}\) for \(i=2,\ldots,n\) and \(N(e_ 1)= \varepsilon e_ n\). Let \(t\) be an integer \(\geq 1\) which is relatively prime to \(n\); we write \(t=rn+s\), where \(r\) is an integer \(\geq 0\) and \(0<s<n\). The \(t\)-th power \(N^ t\) of \(N\) is given by \(N^ t(e_ i)=\varepsilon^ re_{i-s}\) if \(s<i \leq n\), \(N^ t(e_ i)=\varepsilon^{r+1}e_{i-s+n}\) if \(1 \leq i \leq s\). We shall take \(T=N^ t\); this is a nil-elliptic endomorphism. We set \(Y=X^ T\), \(\tilde Y=\tilde X^ T\); as above, we have a morphism \(\rho':\tilde Y\to Y\). Main result:(i) The Euler characteristics of \(Y\), \(\tilde Y\) are given respectively by \((n+t-1)!(n!)^{-1}(t!)^{-1}\), \(t^{n-1}\).(ii) The varieties \(Y\), \(\tilde Y\) can be stratified into finitely many strata each of which is isomorphic to an affine space over \(k\). In particular, their étale cohomology vanishes in odd degrees. Cited in 17 Documents MSC: 14F45 Topological properties in algebraic geometry 14N99 Projective and enumerative algebraic geometry 14F17 Vanishing theorems in algebraic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds 14E99 Birational geometry Keywords:fixed point varieties; stratification; sequences of lattices; flag manifold; endomorphism; Euler characteristics; vanishing of étale cohomology PDFBibTeX XMLCite \textit{G. Lusztig} and \textit{J. M. Smelt}, Bull. Lond. Math. Soc. 23, No. 3, 213--218 (1991; Zbl 0779.14004) Full Text: DOI