dimensions of Galois representations appearing in the cohomology

2011-06-16T13:44:55Z

Let's take $D$ to be $n^2$-dimensional central algebra with an involution $*$ over a CM field $E=FE_0$ (where F is totally real, and $E_0$ is imaginary quadratic). Define $$G(R) = { x \in D \otimes _{\mathbb{Q}} R | x x^* \in R^{\times} }$$ for $\mathbb{Q}$-algebras $R$. Complete this data to a Shimura datum (i.e. define $h: Res _{\mathbb{C} /\mathbb{R} } \mathbb{G} _m \rightarrow G _{\mathbb{R}}$ and $X$) just like in the book of Harris-Taylor, so that we get a Shimura variety $Sh _ K = G(\mathbb{Q}) \backslash G(\mathbb{A} _f) \times X / K$ of unitary type. Choose an irreducible representation $\xi$ of $G(\mathbb{A} _f)$ and form the associated sheaf $\mathcal{L} _{\xi}$ on $Sh _K$. Consider now the cohomology groups of $Sh _K$ and their decomposition: $$H^{n-1} _{et} (Sh_K, \mathcal{L} _{\xi}) = \oplus _{\pi } V _{\pi} \otimes \pi _f ^K$$ where $\pi$ runs over certain automorphic representations (I don't specify which) and where $V _{\pi} = Hom (\pi _f ^K, H^{n-1} _{et} (Sh_K, \mathcal{L} _{\xi}))$ (homomorphisms which are equvariant with respect to Hecke algebra).

Questions: What is known about the dimension of $V_{\pi}$ at the moment? Is it always equal to $n$? If not, what are the other possibilities for $dim V_{\pi}$? Fujiwara in his ICM 2006 talk ("Galois deformations and arithmetic geometry of Shimura varieties"), in footnote 31, says that probably it follows (for $\mathcal{L} _{\xi} = \bar{\mathbb{Q}} _l$ and for cuspidal locus) from the work of Ngo-Laumon on the fundamental lemma. Could someone provide either an explanation or a reference why is that?

Edit: additional question : what can be said of dimension of $V_{\pi}$ which appears in the cohomology group $H^{n-1} _{et} (Sh_K, \mathcal{L} _{\xi}) _{\mathfrak{m}}$ - i.e. localised at some "nice" (for example, non-Eisenstein) maximal ideal $\mathfrak{m}$ of Hecke algebra.

dimensions of Galois representations appearing in the cohomology - MathOverflow

dimensions of Galois representations appearing in the cohomology