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Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$$K$$ $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$$F$$ $F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$$K$$ $K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$$\mathrm{GL}_{n}(F)$$ ${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters. 

The main result of this paper is the uniqueness of local arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. A parametric version implies that the space of germs of symplectomorphisms that preserve the local model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically. This paper can be read independently, but it is part of a series of papers by the authors on the arborealization program. 

Improved data acquisition with autonomous sensor networks for fine-grained data sampling and collection is critical for ensuring urban water sustainability. It leads to better analysis, predictability, and optimization of water resources. To address this need, here, we propose a microwave sensor array composed of five complementary split ring resonators (CSRRs) operating in the frequency range of 1 GHz to 10 GHz. The use of an array provides the opportunity to acquire more information regarding the water pollutants. Here, the design of the proposed sensor array is presented along with the results of testing that with water samples with heavy metal pollutants including Chromium (Cr), Lead (Pb), and Mercury (Hg).