Abstract:Let $X$ be an affine spherical variety, possibly singular, and $\mathsf L^+X$ its arc space. The intersection complex of $\mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and $L$-monoids. In this paper, we compute this intersection complex for the large class of those spherical $G$-varieties whose dual group is equal to $\check G$, and the stalks of its nearby cycles on the horospherical degeneration of $X$. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional $\check G$-representation determined by the set of $B$-invariant valuations on $X$. We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of $\mathsf L^+X$ as a ratio of local $L$-values for a large class of spherical varieties.

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中文翻译：

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交叉路口复合体和未分枝的 𝐿 因子

摘要：令$X$是一个仿射球面变量，可能是单数，$\mathsf L^+X$是它的弧空间。$\mathsf L^+X$ 的交集复形，或者更确切地说是它的有限维形式模型，被推测与局部未分支 $L$-函数的特殊值有关。这种关系以前在 Braverman-Finkelberg-Gaitsgory-Mirković 中建立，用于通过抛物线的单能自由基对还原群的商进行仿射闭合，在 Bouthier-Ngô-Sakellaridis 中建立了复曲面变体和 $L$-monoids。在本文中，我们为那些对偶群等于 $\check G$ 的大类球形 $G$-变种及其附近循环的茎在 $X$ 的日球退化上计算了这个交集复数。我们根据柏原水晶制定答案，它推测对应于由 $X$ 上的 $B$-不变估值集合确定的有限维 $\check G$-表示。我们在很多情况下证明了后一种猜想。在层函数字典下，我们的计算给出了 $\mathsf L^+X$ 的 IC 函数的 Plancherel 密度的公式，它是一大类球形变体的局部 $L$-值的比率。

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