Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. 


Let $X$ be the (complete) flag variety of $G$ dimension $n$, hence we can assume that $X=G/B$. Furthermore let $x \in X$ be the unique fixed point of $B$ under the natural left action of $G$ on $X$. Then denote for $w \in W$ by $X\_w=Bwx$ the corresponding schubert cell in $X$ with dimension $l(w)$.


According to p. 51 in Brylinskis paper "Differential operators on the flag varieties" we have that the formal character of $H^{n-l(w)}\_{X\_w}(X, \mathscr{O})$ is given by $\frac{e^{-w(\rho)-\rho}}{\prod\_{\alpha \in R(B)}(1-e^{-\alpha})}$. Here $R(B)$ is the set the roots of $B$, hence the positive roots of $G$ with respect to $B$ and $\rho$ is the half sum of all positive roots of $G$ with respect to $B$. Finally he concludes that $H^{n-l(w)}\_{X\_w}(X, \mathscr{O})$ has highest weight $-w(\rho)-\rho$. 


He refers to Kempfs "The Grothendieck-Cousin Complex of an Induced Representation" for a proof of this result. But when I look it up there, concretely Lemma 12.8, I would rather say that the formal character of $H^{n-l(w)}\_{X\_w}(X, \mathscr{O})$ is given by $\frac{e^{w(\rho)+\rho}}{\prod\_{\alpha \in R(B)}(1-e^{\alpha})}$. So the sign changed everywhere. Hence it would have lowest weight $w(\rho)+\rho$. 


Can anyone solve my confusion?



Formal character of local cohomology groups with support in Schubert cells
The formula 12.8 in Kempf follows by plugging in 6.5 into 11.10 (see comment by CJS under the question). In the statement of 11.10 Kempf defines $[\chi]$ to denote the isomorphism class of $V(\chi).$ Now $V(\chi)$ is the dual of $\textbf{V}(\chi)$ (see page 313 or 380) and the torus $T$ acts on $\textbf{V}(\chi)$ by $(g,v) \mapsto \chi(g) v$ (see p. 313) which makes the weight/character of $V(\chi)$ to be equal to $-\chi.$ This reconciles Kempf's result/notation with Brylinski's statement.


