We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan--Lusztig $R^x$-polynomials of a pircon $P$ are a $P$-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan--Lusztig theory.

中文翻译：

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Pircon 核和上下对称

我们证明了称为上下对称的对称性质意味着 pircon $P$ 的 Kazhdan--Lusztig $R^x$-多项式是 $P$-核，并且我们证明该性质在经典案例。然后，我们在抛物线 Kazhdan--Lusztig 理论中增强并扩展了 Deodhar 的二元性。