# Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying:

1. $F^i V \supset F^j V$ when $i < j$.
2. $F^i V = V$ for $i << 0$. $F^i V = \{ 0 \}$ for $i >> 0$.
3. $F^i V = \bigcap_{j < i} F^j V$.

We define: $$F^{i+} V = \bigcup_{j > i} F^j V.$$

The slope of $(V,F)$ (when $V \neq \{ 0 \}$) is the rational number: $$M(V,F) = \frac{1}{dim(V)} \sum_{i \in Q} i \cdot dim(F^i V / F^{i+} V).$$

The pair $(V,F)$ is called semistable if $M(W, F_W) \leq M(V, F)$ for every subspace $W \subset V$, with the subspace filtration $F_W$.

A paper of Faltings and Wustholz constructs an additive category with tensor products, whose objects are semistable pairs $(V,F)$. A paper of Fujimori, "On Systems of Linear Inequalities", Bull. Soc. Math. France, seems to imply that the full subcategory of slope-zero objects (together with the zero object) is Tannakian (the abelian category axioms require semistability), with fibre functor to the category of $k$-vector spaces (though Fujimori considers quite a bit more).

Does anyone know another good reference for the properties of this Tannakian category? Can you describe the associated affine group scheme over $k$? I'm particularly interested, when $k$ is a finite field or a local field.

Update: I think the slope-zero requirement is too strong (though it is assumed in Fujimori). It seems to exclude almost all the semistable pairs $(V,F)$, if my linear algebra is correct. Anyone want to explain this to me too?

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