A limit theorem for torus translation
In this article, we prove a limit theorem for an observable on the torus with a singularity of type $x^{-a}$, where $0<a<1$.
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arXiv subjects
Linyuan Liu
Publications and source records attributed to Linyuan Liu.
Cohomologie des fibrés en droites sur SL3 /B en caractéristique positive : deux filtrations et conséquences
In this thesis, I will prove the existence of two filtrations of the cohomology of line bundles on SL_3/B. The first one is a two-step filtration that exists for $H^1(μ)$ and $H^2(μ)$ if $μ$ is in the Griffith region. The second one exists for all $H^i(μ)$, which is similar to the p-filtration that has been considered by Jens Carsten Jantzen.
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On the cohomology of line bundles over certain flag schemes II
Over a field $K$ of characteristic $p$, let $Z$ be the incidence variety in $\mathbb{P}^d \times (\mathbb{P}^d)^*$ and let $\mathcal{L}$ be the restriction to $Z$ of the line bundle $\mathcal{O}(-n-d) \boxtimes \mathcal{O}(n)$, where $n = p+f$ with $0 \leq f \leq p-2$. We prove that $H^d(Z,\mathcal{L})$ is the simple $\operatorname{GL}_{d+1}$-module corresponding to the partition $λ_0 = (p-1+f,p-1,f+1)$. When $f= 0$, using the first author's description of $H^d(Z,\mathcal{L})$ and Jantzen's sum formula, we obtain as a by-product that the sum of the monomial symmetric functions $m_λ$, for all partitions $λ$ of $2p-1$ less than $(p-1,p-1,1)$ in the dominance order, is the alternating sum of the Schur functions $S_{p-1,p-1-i,1^{i+1}}$ for $i=0,\dots,p-2$.
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On the cohomology of line bundles over certain flag schemes
Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $α_{2},\cdots,α_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags $\{D_{1}\subset H_{d}\subset V\}$. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.
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Filtrations of tilting modules and costalks of parity sheaves
Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.
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