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The study of certain union $$X(\mu , b)$$ of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for $$X(\mu , b)$$ associated to sufficiently large dominant coweight $$\mu $$ .

The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties. We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge–Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic $$\sigma $$ -isogeny class is Hodge–Newton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152(3–4):317–343, 2017) this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.

The Newton strata of a reductive p-adic group are introduced in He (Forum Math Pi 6:e2, 2018) and play some role in the representation theory of p-adic groups. In this paper, we give a geometric interpretation of the Newton strata.

In the paper four stratifications in the reduction modulo p of a general Shimura variety are studied: the Newton stratification, the Kottwitz–Rapoport stratification, the Ekedahl–Oort stratification and the Ekedahl–Kottwitz–Oort–Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case.

The cocenter of an affine Hecke algebra plays an important role in the study of representations of the affine Hecke algebra and the geometry of affine Deligne–Lusztig varieties (see for example, He and Nie in Compos Math 150(11):1903–1927, 2014; He in Ann Math 179:367–404, 2014; Ciubotaru and He in Cocenter and representations of affine Hecke algebras, 2014). In this paper, we give a Bernstein–Lusztig type presentation of the cocenter. We also obtain a comparison theorem between the class polynomials of the affine Hecke algebra and those of its parabolic subalgebras, which is an algebraic analog of the Hodge–Newton decomposition theorem for affine Deligne–Lusztig varieties. As a consequence, we present a new proof of the emptiness pattern of affine Deligne–Lusztig varieties (Gortz et al. in Compos Math 146(5):1339–1382, 2010; Gortz et al. in Ann Sci Ecole Norm Sup, 2012).

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

Let $G$ be a semisimple adjoint group over $\bold C$ and $\bar{G}$ be the De Concini-Procesi completion of $G$. In this paper, we define a Lagrangian subvariety $\Lambda$ of the cotangent bundle of $\bar{G}$ such that the singular support of any character sheaf on $\bar{G}$ is contained in $\Lambda$., Comment: 8 pages

An increased ratio between serum levels of immunoreactive anionic and cationic trypsin is a common finding in many forms of pancreatic disease. Experimental studies have shown that increased stimulation with cholecystokinin (CCK) leads to an increase in the ratio between the pancreatic content of anionic and cationic trypsin. To study whether this effect is caused by increased pancreatic synthesis of anionic trypsin in relation to cationic trypsin we studied the levels of mRNA for anionic and cationic trypsinogen in pancreatic extracts from rats exposed to increased CCK levels through chronic subcutaneous administration of CCK. Northern blot and slot blot hybridization techniques were used. The ratio between mRNAs for anionic and cationic trypsin was significantly higher in CCK-treated rats: median level, 2.04 (range, 1.33-4.08) versus a median level of 1.15 (range, 0.97-2.17) in the control group: P0.01. These findings support the view that chronic CCK stimulation leads to the increased synthesis of anionic trypsinogen compared to cationic trypsinogen.