A Gröbner basis for Kazhdan-Lusztig ideals

{\it Kazhdan-Lusztig ideals}, a family of generalized determinantal ideals investigated in [Woo-Yong~'08], provide an explicit choice of coordinates and equations encoding a neighborhood of a torus-fixed point of a Schubert variety on a type $A$ flag variety. Our main result is a Gr\"{o}bner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the {\it Rothe diagram of a permutation}, and their appearance in: \begin{itemize} \item combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomialsof [Lascoux-Sch\"{u}tzenberger '82]; \item the equivariant $K$-theory specialization formula of [Buch-Rim\'{a}nyi '04]; and \item a positive combinatorial formula for multiplicities of Schubert varieties in good cases, including those for which the associated Kazhdan-Lusztig ideal is homogeneous under the standard grading. \end{itemize} Our results generalize (with alternate proofs) [Knutson-Miller '05]'s Gr\"{o}bner basis theorem for Schubert determinantal ideals and their geometric interpretation of the monomial positivity of Schubert polynomials. We also complement recent work of [Knutson '08 $\&$ '09] on degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian multiplicity formulas.