ENDOSCOPIC CLASSIFICATION OF VERY CUSPIDAL REPRESENTATIONS OF QUASI-SPLIT UNITARY GROUPS

Transcription

1 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA REPRESENTATIONS OF QUASI-SPIT UNITARY GROUPS GEO KAM-FAI TAM Abstract. We describe the supercuspidal representations within certain stable packets, classified by Arthur and Mok using the theory of endoscopy, of an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The description is given in terms of types constructed by Bushnell- Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy some regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder. Our main result is analogous to the essentially tame local anglands correspondence of Bushnell-Henniart for a general linear group: we need to correct the types by twisting by certain characters, called amending characters in this paper, in order to classify these representations into stable packets. Contents 1. Introduction Endoscopic classification Very cuspidal representations Methodology Remarks related to the literature Acknowledgements 8 2. Unitary groups Notations Unitary groups and subgroups Embeddings of tori attice sequences Hereditary orders and compact subgroups attices of higher ranks Descriptions on filtration subgroups Conjugate actions on unipotent subgroups Representations Characters Maximal types for general linear groups Cuspidal types for unitary groups Covering pairs Hecke algebras Structures of Hecke algebras Points of reducibility Reduction to finite reductive quotients Amending Preliminary on group theory 40 1

2 2 GEO KAM-FAI TAM 5.2. Amending characters The amending process The product of amending characters Values of amending characters Endoscopic classification groups and parameters Asai -functions Extended cuspidal supports of very cuspidal representations Stable packets Base change in general 56 References Introduction As mentioned in the abstract, we describe in this paper the supercuspidal representations within certain endoscopic stable packets of an unramified quasi-split unitary group over a p-adic field of odd residual characteristic, and this description is given by types. These packets illustrate a relation between the theory of endoscopy and the theory of types, both of which are of fundamental interest. As a starting example, we focus on stable packets consisting of very cuspidal representations. We give an overview of our strategy towards the description of our stable packets of an unramified quasi-split unitary group G. We first interpret the endoscopic classification by Arthur [4] and Mok [50] for G as classifying a finite collection of supercuspidal representations of some general linear groups, called the extended cuspidal support in Mœglin s theory of base change [48], associated to a given packet. This can be reinterpreted as another problem, again by Mœglin s theory, of the reducibility of a parabolically induced representation of a larger unitary group from a supercuspidal representation of a evi subgroup of the form G n G. We hence construct, via the theories of Bushnell-Kutzko [9, 14], Stevens [66], and Miyauchi-Stevens [45], a covering type for such a parabolic induction and study its Hecke algebra, whose structure can be described using a theorem of usztig in [44] and is related to the above reducibility by a theorem of Blondel [7]. Finally, we apply certain facts, due to Mœglin [48] and Henniart [31], about -functions defined by the anglands-shahidi method [61] to determine the correct representations in the extended cuspidal support. In this Introduction section, we elaborate the above strategy. We first recall some basic notions from the two theories: endoscopic classification, and the construction of supercuspidal representations induced from types. We then state our main Theorem 1.1, which summarizes the two detailed theorems (Theorems 5.9 and 6.7) of this paper. After that, we detail briefly the methodology for studying the Hecke algebras and proving the theorems. Finally, we provide several remarks related to the literature Endoscopic classification. et F be a p-adic field whose residual characteristic is odd, and let F/F be a field extension, either trivial or quadratic. et G be a quasi-split connected classical group over F, which is either a special orthogonal group or a symplectic group when F = F, or a unitary group when F/F is

3 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA... 3 quadratic. The irreducible tempered representations of G = G(F ) and their stable packets (sometimes called -packets or simply packets) have been classified by [4, Theorem 1.5.1] and [50, Theorem 2.5.1] using the theory of endoscopy, among many important results. In this paper, we only focus on unitary groups. We hence have G = G V = U n (F/F ), a unitary group defined by a Hermitian space V where dim F V = n. Under the endoscopic classification, the discrete packets (those consist of discrete series representations) can be parametrized by n-dimensional complex representations of the Weil-Deligne group WD F := W F SU 2 of F, whose components are conjugate-self-dual and occur without multiplicity. We call these representations discrete parameters. The classification of discrete packets can be described more explicitly using [48, 49]. Suppose that φ is a discrete parameter and Π φ is the corresponding packet, then the extended cuspidal support of each representation in Π φ coincides with the cuspidal support of the representation π φ corresponding to φ under the local anglands correspondence for G n [28, 30, 59]. We follow the terminology in [48] and call π φ the base change of Π φ. For the precise notion of extended cuspidal support, we refer to [48] to avoid further details, but we provide an example which will be adopted throughout our paper. We assume that φ is trivial on SU 2. By the definition in [48, 5.3], that a supercuspidal representation π of G n (F ) is in the extended cuspidal support of a representation π of G V is equivalent to that the normalized parabolically induced representation (1.1) π det s π := i G W P ( π det s π), s C, is reducible at s = 1. Here det stands for the character x det x F for x G n (F ), G W is the unitary group defined by the Hermitian space W which is the orthogonal sum of V and a hyperbolic Hermitian space of dimension n, and P is a parabolic subgroup of G W whose evi component M is isomorphic to G n (F ) G. Hence π Π φ if and only if π lies in the cuspidal support of π φ, and if and only if the anglands parameter of π is one of the components of φ. The reducibility of (1.1) for quasi-split unitary groups was studied in [48], and analogously for symplectic and special orthogonal groups in [47], some of whose ideas was originated from [63, 61]. This kind of reducibilities is related to the poles of certain intertwining operators, given in terms of anglands-shahidi -functions [61]. The goal of this paper is to describe the representations in Π φ, where this description will be made more precise in the next section. As a starting example, we only focus on the following parameters in this paper. We require that F/F is unramified and φ is a finite sum of conjugate-self-dual irreducible components: (1.2) φ = φ i, φi = Ind W F W ξi Ei, i I where E i /F is an unramified extension (necessarily of odd degree by conjugateself-duality), and ξ i is a character of E i, viewed as a character of W E i by Artin reciprocity. We further require that all characters { ξ i } i I satisfy certain regularity conditions (which will be detailed in Section 3.1.1) and, for simplicity, have a common level d.

4 4 GEO KAM-FAI TAM When d = 0, these parameters are examples of the depth-zero parameters in [19] for general unramified reductive groups. In this case, for odd special orthogonal or symplectic groups, the extended cuspidal supports of the corresponding representations can be determined by the results in [58]. When d is positive, these parameters are examples of those in [55], whose corresponding representations are called very cuspidal representations. We follow this terminology and call φ a very cuspidal parameter Very cuspidal representations. In this section, we summarize the construction of types for very cuspidal representations of general linear groups and quasi-split unitary groups. Then we will state our main result, Theorem 1.1. We first summarize the construction for a general linear group. Suppose that φ is a parameter appearing as one of the components in the parameter (1.2), which takes the form Ind W F W ξ E. By regarding E as an unramified elliptic maximal torus of G n (F ) where n = [E : F ], we construct in Section 3.2 a finite dimensional representation, called a maximal simple type in [9, (5.2)], of a compact open subgroup using the character ξ. The maximal type is then extended to an open compact-mod-center subgroup and induced to a supercuspidal representation π ξ of G n (F ) such that its anglands parameter is φ. When G = U n,f/f is an unramified quasi-split unitary group, we know that every unramified elliptic maximal torus of G is of the form i I U 1,E i/e i, where E i /F is an unramified extension of degree n i (necessarily odd), E i /E i is quadratic unramified, and i I n i = n. et T be a torus of this form. In contrast to the G n -case, the F -embeddings of T in G are in general not conjugate to each other by G = G(F ). We use an index x to stand for a G-conjugacy class of an embedding of T = T(F ) in G, and denote the image in G (up to G-conjugacy) by T x. We briefly describe the construction in Section 3.3 of very cuspidal representations of G. Given a set of characters { ξ i } i I as in 1.2, we assume for now that each ξ i is a +-skew character, which means that ξ i E i is trivial. Write ξ := i I ξi the product character of i I E i. By descending each ξ i to a character ξ i of U 1,Ei/E i canonically, we form another product character ξ := i I ξ i of T x. We hence construct from ξ a maximal semi-simple type in the sense of [66], which is a cuspidal type as defined in [45]. This type is then compactly-induced to a supercuspidal representation π x, ξ of G. To be more precise, our construction of π x, ξ is a combination of the methods in [66] and [1]: we construct from ξ a character, called a semi-simple character in [65], of a compact subgroup of G using [1] (see also [55, Section 3.2]), and extend this character to a cuspidal type using the idea of beta-extensions in [66, Section 4]. We emphasize that it is important to consider different conjugacy classes of F -embeddings x of T in G. From many examples (e.g. the depth-0 case [19]) it is known that different embeddings give rise to non-isomorphic supercuspidal representations π x, ξ, for a fixed ξ. With the descriptions of the above supercuspidal representations, we now state our main result, summarizing Theorems 5.9 and 6.7 in a simpler way. Theorem 1.1. et n be an odd positive integer, φ be a very cuspidal parameter, and ξ be the character of i I E i associated to φ. For each G-conjugacy class x

5 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA... 5 of embeddings of T, there exists a unique character ν y of x, ξ i I E i or quadratic (hence tamely ramified), such that Π φ = {π x, ξ ν y } x. x, ξ, either trivial In other words, the base change of the above supercuspidal representations is π ξ, the representation of G n (F ) whose cuspidal support is { π ξi } i I. We remark that Theorem 1.1 also holds for even n, except that we need to modify ξ by multiplying to it a character related to an endoscopic datum. This will be explained in Section 6. We call ν y x, ξ the amending character associated to (x, ξ). In the next section, we will explain the notation y for the character ν y and provide a brief idea of x, ξ computing it, together with the methodology for proving Theorem Methodology. Using Mœglin s theory mentioned in Section 1.1, our work is to determine which data ( ξ, x, ξ) give rise to the reducibility of (1.3) π ξ det s π x, ξ at s = 1. We first use a result of [7]: the real-parts of all s at which (1.3) is reducible are determined by the structure of a Hecke algebra of a (non-cuspidal) type in the larger unitary group G W, which covers the (exterior) product of the maximal simple type and the cuspidal type constructed in Section 1.2 for π π ξ x, ξ. The notion of covering types is defined in [14] for connected reductive groups. In terms of category theory, the category of non-degenerate right modules of the above Hecke algebra determines a Bernstein component, the full-subcategory of smooth representations of G W whose irreducible subquotients have supports in the inertial class of π π ξ x, ξ. We then combine the machineries of [14, 66, 7] (and also [51] for the depth-zero situation) in Sections 3 and 4 to determine the structure of the aforementioned Hecke algebra, which is reduced to considering certain Hecke algebras for unitary groups over finite fields, whose structures are given by a theorem of usztig [44, Theorem 8.6]. The Hecke algebra turns out to be either isomorphic to the polynomial ring in one invertible variable, or a Hecke algebra on an infinite dihedral group with two invertible generators, each satisfying a quadratic relation. The covering type that gives rise to the latter Hecke algebra is the one corresponding to the inertial class of that provides the reducibility of (1.3) at s = 1, in which case π ξ the calculation implies that E is one of the E i, and the two characters ξ and ξ i differ from each other by a character ν y i = νy whose order is at most 2 (hence is i,x, ξ tamely ramified). The values of ν y i on o E i will be computed in Section 5.5. Actually, this is the signature character of the conjugate action of o E i on a finite quotient of two compact open subgroups of G W, which can be viewed as Moy-Prasad filtration subgroups [52]. These compact subgroups will be described using the language of lattice sequences and hereditary orders from [66], as we will use his results heavily in our calculation. The notation y of the character ν y i stands for a hyperspecial vertex in an appropriate Bruhat-Tits building, so that we obtain a finite unitary group of the

6 6 GEO KAM-FAI TAM largest possible rank in order to apply usztig s theorem to determine the structures of the Hecke algebras. Now the inertial class of is determined. Within this class there are two π ξ conjugate-self-dual representations, the possible candidates for the reducibility of (1.3) at s = 1, different from each other by the quadratic unramified character of G n (F ). An application on the equality between Asai -functions of parameters and those defined by the anglands-shahidi method [61], proved by Henniart [31], determines the one that gives rise to the reducibility, and hence determines ν y i completely. Finally, when n is odd (and under a slight modification when n is even) we set ν y = x, ξ i I νy i and obtain our amending character in Theorem 1.1. We hence obtain a collection of conjugate-self-dual supercuspidal representations which belong to the extended cuspidal support of Π φ. By applying Mœglin s inequality [48, 4. Proposition] on estimating the ranks of the underlying unitary groups, we claim that our collection exhausts the entire extended cuspidal support. This completes our methodology of proving Theorem 1.1. We provide a few remarks related to our methodology. (i) A similar kind of amending characters already appeared in [7, Section 3.3], where she studied the reducibility of (1.3) in the disjoint case; in our language, this means that ξ is not Galois conjugate to any ξ i for i I. This is reduced to study the reducibility of the parabolically induced representation of U 2n from det s regarded as a representation of the Siegel parabolic π ξ subgroup containing the evi subgroup isomorphic to G n. This kind of parabolically induced representations was previously studied in [23, 24, 54] for unitary groups, in [62, 53] for split classical groups, and in [25] for arbitrary classical groups using the theory of covering types. (ii) If z is a vertex adjacent to y, we can define another amending character ν z = x, ξ i I νz i analogously for z in place of y. We describe in Proposition 5.7 a relation between our amending characters ν y i and ν i z with a character introduced in [66, Corollary 6.13]. In our paper, this character is denoted by χ z y,i defined on o E i, and is called the transition character between betaextensions relative to the parahoric subgroups associated to y and z. Our Proposition 5.7 simply says that χ z y,i = ( ν y i νz i ) o E i. A similar relation between these characters is also known in the Siegel case [7] (see the discussion before and in Proposition 3.17 loc. cit.). (iii) When using the Asai -functions to pick out the correct representations for the reducibility in (1.3), we view G as an elliptic twisted endoscopic group of G = Res F/F G n. In general, an elliptic twisted endoscopic datum of G is of the form H = U n1,f/f U n2,f/f, n 1, n 2 Z 0 and n 1 + n 2 = n, attached with a choice among two sign-data that determines the embedding of the -group H into G (except when n1 = n 2, in which case the two data are equivalent in an appropriate sense). We can also study the base change

7 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA... 7 of packets of H(F ) to representations of G(F ) = G n (F ). In Corollary 6.10, we generalize Theorem 1.1 to a statement for arbitrary elliptic twisted endoscopic group H of G, so that the Theorem is viewed as the case H = G Remarks related to the literature. (i) The quasi-split condition for a unitary group is applied because, in order to describe the base change, we view the group as a twisted endoscopic group of a general linear group. This is the view-point in [48, 4, 50]. However, as pointed out in [49], this condition is only for simplicity. Now the endoscopic classification for inner forms of classical groups is available (the essential idea is in [4, Chapter 9] for special orthogonal and symplectic groups, and in [36] for unitary groups). The author expects that the same method in this paper applies to describe the local transfers (whenever defined) for inner forms without significant modification. (ii) The above constructions of extended maximal simple types and cuspidal types from characters of elliptic maximal tori are well-known, where the origin can be traced back to [33] for general linear groups (and similar constructions in [22, 16]). The method was then generalized to [9], where we allow the underlying types to be directly extended from characters of compact subgroups without using characters of elliptic maximal tori, and hence obtain an exhaustive construction. Similar exhaustive methods were then applied to special linear groups [13] and to classical groups [66, 45] (when the residual characteristic is odd). There are other similar methods [1, 69, 37] and recently some novel methods [26, 56] aiming at constructing supercuspidal representations for general connected reductive groups. (iii) Attempts to describing local transfers include the depth-zero case, for G = U 2 and U 3, by [2] when F/F is unramified, and by [3] when F/F is ramified; the positive level case, by [6] for G = U 2, and by [5] for G = U 3 when the packet is a singleton. In the cases above, they directly computed the representations in a packet by applying the twisted trace formula [57]. For using extended cuspidal support, we have the following results. For G being odd special orthogonal or symplectic, [58] described the local transfers for depth-zero parameters from [19], and proved that the local transfers are the same as in [34, 18]; [42] adopted the same method for depth-zero GSP 4 and proved that the local transfers are the same as in [21]. It seems to the author that the present paper is the first place to apply the method of extended cuspidal support for positive level supercuspidal representations. (iv) Some properties of our amending characters resemble those of the rectifiers introduced in [10, 11, 12], for example, both characters can be expressed as signature characters of the toral actions on the root spaces in the level of reductive quotients; our amending characters emerge from endoscopic classification, while the rectifiers emerge from automorphic induction [32], which can be regarded as an example of twisted endoscopy.

8 8 GEO KAM-FAI TAM In the previous work of the author [67], he expressed the rectifiers in terms of certain endoscopic data and admissible embeddings of -tori. He expects that our amending characters have a similar expression, and hopes to address this issue in his later work. Finally, there is a recent result [35] on the relation between admissible embeddings of -tori (called an -embedding in loc. cit.) with Reeder-Yu s epipelagic supercuspidal representations [56] for general tamely ramified reductive groups, using the approach of endoscopic character formulas. (v) We discuss the generality about our methodology towards describing supercuspidal representations and discrete series for quasi-split classical groups. As we have seen, we only study the endoscopic classification for representations of quasi-split unitary groups under the restrictive very cuspidal condition above. The purpose of imposing this condition is to take advantage of the simplicity of the parameters and determine the values of the amending characters in Proposition For parameters whose components are essentially tame (in the sense of [10]), the values of their amending characters depend heavily on the parities of jumps which obey certain special patterns. The complete description of these values is given case-by-case and too complicated for a starting example. The author hopes to deal with this in later work. Other than this, he expects that many parts of the present paper can be either applied to the general situation directly, or else generalized easily. Furthermore, in an ongoing joint work (now completed as in [8]) of Blondel, Henniart, and Stevens, they had obtained some results for symplectic groups on the relation between covering types and packets of representations whose underlying cuspidal types are not necessarily constructed from characters. When combined with endoscopic classification, their results should lead to describing explicitly the local transfers of packets for arbitrary classical groups Acknowledgements. This project was started when the author was a postdoc in Jussieu, receiving funding from the European Research Council under the European Communitys Seventh Framework Programme (FP7/ )/ERC Grant agreement no (AAMOT). The paper was then written when he was a postdoc in McMaster University. He would like to thank Jim Arthur, Anne-Marie Aubert, Kei-Yuen Chan, Michael Harris, Guy Henniart, Chung-Pang Mok, Fiona Murnaghan, Freydoon Shahidi, and Bin Xu for discussions. He would like to especially thank aure Blasco, Corinne Blondel, Robert Kurinczuk, Colette Mœglin, Shaun Stevens, and Jean-oup Waldspurger for explaining their papers in detail. He would like to thank Charles i and Manfred Kolster for their supports and encouragements. The present work was greatly inspired by the discussion with Shaun Stevens during the author visiting UEA in March 2013, when he explained to the author the relation between reducibility and stable packets, and the method of using covers and Hecke algebra to compute reducibility points in his joint work with Corinne Blondel and Guy Henniart. The author had useful exchanges with Corinne Blondel from April 2015, who made him available the slides of her talk in uminy in January 2014, giving more details on these methods. The author is indebted to both mathematicians. During the peer-review process, the author received manuscripts from Corinne Blondel on the work in progress of Blondel, Henniart, and Stevens (an

9 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA... 9 earlier version of [8]), and the recent work of ust and Stevens on the depth-zero case [43]. The author was substantially benefited from these manuscripts. ast but not least, he would like to thank the referees for their careful readings. 2. Unitary groups 2.1. Notations. et F be a p-adic field of odd residual characteristic p, and F/F be the unramified quadratic extension. We denote by o F and p F (resp. o F and p F ) the ring of integers and its maximal ideal of F (resp. F ). We choose a uniformizer ϖ F such that ϖo F = p F. We denote by k F (resp. k F ) the residual field of F (resp. F ) with cardinality q (resp. q). The multiplicative group F decomposes into a product of subgroups ϖ µ F U 1 F. These subgroups are respectively the group generated by the uniformizer ϖ, the group µ F of order q 1 containing the roots of unity of order prime to p, and the 1-unit group UF 1 := 1 + p F. We identify µ F with k F by the natural projection. We have a filtration of fractional ideals p n F = ϖ n o F in F for all integers n, and a filtration of subgroups UF n := 1 + p n F of U F = o F for positive integers n. It will be convenient to extend the notation and define UF r = U r F for r R 0, where r is the smallest integer r. We also define U r+ F = s>r UF s. Given a subgroup G of F containing UF 1, we call a character χ of G tamely ramified if it is trivial on UF 1. We fix, once and for all, an additive character ψ of F whose conductor is p, so that its composition with the trace tr F/F is an additive character ψ F of F whose conductor is p F. We denote by δ F/F the character associated to the quadratic extension F/F by local class field theory, which is the quadratic character of F trivial on the norm group N F/F (F ). We denote by Γ F the absolute Galois group of F and by c a Frobenius element in Γ F such that c x x q mod p F, for all x o F. We denote the non-trivial element in Γ F/F = Γ F /Γ F and that in Γ kf /k F = Γ kf /Γ kf also by c. We provide some notations on general group and representation theories. If G acts on a set X and X is a subset of X, we denote by N G (X ) the subgroup of elements in G that leave X invariant, and by Z G (X ) the subgroup of elements in G that act trivially on X. If B and C are subrings of a ring A, we denote by Z B (C) to be the elements in B which commute with every element in C. Given a representation π of G, we denoted by π the contragradient of π. If π and π are representations of two groups G and G respectively, we denote by π π their exterior tensor product, which is a representation of G G. If π and π are respectively representations of two subgroups H and H of G, we denote by I G (π, π ) the subset of elements in G that intertwines π and π, and by I G (π) when H = H and π = π. Suppose that G is a locally profinite group with a countable base of open sets, and K is a compact open subgroup of G. We assume that G is unimodular with a Haar measure. Given a representation λ of K on a finite dimensional C-vector

10 10 GEO KAM-FAI TAM space W, we denote by H(G, λ) the associated Hecke algebra, which is the space of compactly supported functions f : G End C (W ) satisfying f(k 1 gk 2 ) = λ (k 1 ) f(g) λ (k 2 ), for all k 1, k 2 K and g G, with an associative C-algebra structure under the convolution f 1 f 2 (g) = f 1 (x)f 2 (x 1 g)dx, for all g G. G Suppose that δ is a character of a group G. We say that δ is quadratic if its order is exactly 2, and is at most quadratic if its order is 1 or Unitary groups and subgroups. et V be a vector space of F -dimension n, equipped with a non-degenerate Hermitian form h V. The group G V of isometries is the unitary group U n (F/F ), which is the subgroup of F -points of an algebraic group G V = U n,f/f over F, an F -form of the split group G n,f. Explicitly, if we set (2.1) J = J V = we let Γ F act on G V (2.2) ( 1) n , by conjugating the entries and c Γ F Γ F act by σ x = J c ( t x 1 )J 1, for all x G V where c x is the c-conjugate on all entries of x, and t x is the transpose of a matrix x, then G V (F ) = G V. et G V be the algebraic group Res F/F G n,f over F, where c Γ F Γ F acts as (x, y) ( σ y, σ x), for all x, y G V. The subgroup of F -points is G V = G V (F ) = G V (F ) = G n (F ). The action (2.2) then descends to an involution σ on G V, and ( G V ) σ = G V. Note that if dim V = 1, then G V = ker NF/F, which decomposes into a product of subgroups µ F/F U 1 F/F, where µ F/F is the subgroup of µ F of order q +1, and U 1 F/F := {x U 1 F, σ x = x}. Note that we can identify µ F/F with U 1 (k F /k F ) = {x k F, σ x := c x 1 = x} by the natural projection. We denote by ÃV = End F (V ) the algebra of F -endomorphisms of V. Define an action of Γ F/F on ÃV by (2.3) σ X = J c ( t X)J 1 for X ÃV, and write A V = (ÃV ) σ. In this paper, we only consider quasi-split unitary groups defined over F, so G V contains a Borel subgroup invariant under the action of Γ F. From the chosen matrix J, the Borel subgroup is the group of upper triangular unitary matrices.

11 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA Suppose that V admits a decomposition V = i I V i, where I is a finite index set, then we have an embedding (2.4) G Vi G V i I as a evi subgroup defined over F. If such embedding is defined over F, then it restricts to an embedding G i I Vi G V, and all such embeddings are conjugate by G V. We now require moreover that the decomposition V = i I V i is orthogonal with respect to h V, and denote by h Vi the restriction of h V on each V i. The involution σ of G V restricts to the corresponding involution σ i on each G Vi, so that (2.4) restricts to an F -embedding G Vi G V i I and also an embedding i I G V i G V of F -points. However, different F - embeddings may not be conjugate to each other under G V, and there may not be any F -parabolic subgroup containing this evi subgroup. We will also consider some unitary groups of higher ranks and their subgroups. Fix i I and let V i, and V i,+ be two vector spaces isomorphic to V i, then write (2.5) W = W i = V i, V V i,+. By choosing a basis B = {e 1,..., e ni } for V i, and another B + = {e 1,..., e ni } for V i,+, we equip W with a Hermitian form h W defined such that (2.6) V (V i, V i,+) with respect to h W, h W V V = h V, h W (e i, e j ) = h W (e i, e j ) = 0, and h W (e i, e j ) = ϖδ ij, for i, j = 1,..., n i. For convenience, we still denote the involution on G W by σ and let G W = ( G W ) σ be the corresponding unitary group. We denote by P the parabolic subgroup of G W stabilizing the flag (2.7) 0 V i, V i, V W, and by M the evi subgroup of P stabilizing each summand V i,, V, and V i,+ of W. If we write G Vi = G F (V i, ) (or G F (V i,+)), then M = G Vi G V given by a diagonal embedding I M : G Vi G V G W, (g, h) ( σi g, h, g). We also denote by U the unipotent radical of P, and by U the opposite of U Embeddings of tori. et I be a finite set, and for i I, let E i be the unramified extension of F of degree n i, such that E i := E i F F is the unramified extension of F of the same degree. Hence each n i is necessarily odd. We denote by c i the non-trivial automorphism in Γ Ei/E i. We can regard each E i as a Hermitian space by defining h Ei (w i, w i) = tr Ei/F ( ci w i w i), for all w i, w i E i.

12 12 GEO KAM-FAI TAM Suppose that V admits a decomposition V = i I V i such that dim V i = n i. By identifying V i with E i as an F -vector space, we have an F -embedding of the unramified elliptic maximal torus T i = Res Ei/F G 1,Ei G Vi, and hence an F -embedding of the unramified maximal torus T = i I T i G V using the embedding (2.4). We hence have an embedding T i = T i (F ) G Vi each i I, and hence an embedding T = T i G V, i I for by restricting to subgroups of F -points. For convenience, we still denote the image by T = T i I i. Now suppose that the decomposition V = i I V i is moreover orthogonal with respect to h V, and such that each isomorphism of V i with E i is compatible with their Hermitian forms h Vi and h Ei. Then T i is embedded into G Vi such that σ i acts on each factor T i as t i ci t 1 i, for t i T i. There is an F -embedding of T i = Res E i /F U 1,Ei/E i into G Vi for each i I, and hence an F -embedding (2.8) T = i I T i G V as an elliptic unramified maximal torus, and every such torus is of this form. By restricting to subgroups of F -points, we have the corresponding embedding of each T i into G Vi, and an embedding T = T σ = i I T i G V. An embedding I : T G V considered above is defined over F, which means that I γ T = γ GV I, for all γ Γ F, where γ T (resp. γ GV ) is the automorphism of T (resp. G V ) defined by γ. et I 0 be the F -embedding T G V defined in (2.8) with image T 0, and let I : T G V be another F -embedding, which is necessarily conjugate under G V = G V ( F ), i.e., there is g G V such that g 1 T 0 (F )g = I(T)(F ). For any γ Γ F and t T(F ), the relation γ (g 1 I 0 (t)g) = γ g 1 I 0 (t) γ g = γ I(t) = I(t) = g 1 I 0 (t)g implies that g( γ g 1 ) T 0. This defines a cocycle Γ F T whose class lies in D(G V, T, F ) := ker(h 1 (Γ F, T) (I0) H 1 (Γ F, G V )). Proposition 2.1. The set of G V -conjugacy classes of F -embeddings of T in G V within the G V -conjugacy class is bijectively parameterized by D(G V, T, F ). Proof. This follows from simple arguments, or one can see [40, p.702] for details. From now on we abbreviate D(G V, T, F ) by D, and call an F -embedding of T in G V just an embedding. For computation, we provide a well-known description of D.

13 Proposition 2.2. ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA (i) There are isomorphisms H 1 (Γ F, G V ) = Z/2 and H 1 (Γ F, T) = (Z/2) #I. (ii) We can parametrize the set D by the group {(e i ) i I (Z/2) #I, where i I e i = 0} such that I 0 is identified with (0) i I, the zero in (Z/2) #I. Proof. The first isomorphism of (i) is similar to [57, (3.8.1)], where he proved that H 1 (Γ F, U 2,F/F ) = F /N F/F (F ). The second isomorphism is just a special case of the first when dim F V = 1, using Shapiro s emma. For (ii), see [57, Proposition 3.5.2(c)]. Corollary 2.3. We can parametrize D by partitions of I into two subsets I = I o I e such that the cardinality of one of the subsets (say I e ) is even. Proof. This is clear from the description of D in Proposition 2.2. We can explicitly describe the above (non-canonical) parametrization of D using the idea in [68, Section 1.7]. et {E i } i I be the set of field extensions as in the beginning of this section, and fix a tuple x = (x i ) i I, where each x i E i, such that (2.9) δ Ei/E i (x i ) = 1. i I Define a Hermitian form on each E i by h Ei (w i, w i) = tr Ei/F (x i c i w i w i), for all w i, w i E i, and denote by I x the embedding of T into G V using the same procedure as defining I 0 (corresponding to the tuple x with all x i = 1). If T x is the image of I x in G V and T x = T x (F ), then the arguments in loc. cit. show that, for two tuples x and x, the tori T x and T x are G V -conjugate if and only if δ Ei/E i (x i ) = δ Ei/E i (x i) for all i I. Hence we can, and do, choose x i {1, ϖ} to represent the G V -conjugacy class of I x. If we define I o (resp. I e ) to be the subset consisting of i I such that x i = 1 (resp. x i = ϖ), then condition (2.9) implies that #I e must be even. Finally, as T = T(F ), we denote the image I x ( T ) = (I x (T))(F ) in G V = G V (F ) by T x, for x D. For r R 0, if we denote T r i = U r E i and T r = i I T r i and also T r i = U 1 (E i /E i ) U r E i and T r = i I T r i, then we denote the images T r x = I x ( T r ), T r i,x = I x ( T r i ), T r x = I x (T r ), and T r i,x = I x (T r i ).

14 14 GEO KAM-FAI TAM T r+ We also define i, T r+, s = s>r T i, and so on. T r+ i T r+ x, r+ T i,x, T r+ i, T r+, Tx r+, and T r+ i,x by, for example, 2.4. attice sequences. et Λ be an o F -lattice sequence in V, defined as a function Λ from Z to the set of o F -lattices in V such that (i) Λ(k) Λ(j) if j k; (ii) there exists a positive integer e Λ = e(λ/o F ) such that ϖλ(k) = Λ(k + e Λ ) for all k Z. If Λ(k) Λ(j) for all j k, we call Λ a lattice chain. The integer e Λ is called the o F -period, or just period, of Λ. We then extend the domain of Λ from Z to R by defining Λ(r) = Λ( r ) for all r R, and also write Λ(r+) = s>r Λ(s). Recall that h V is an Hermitian form on V that defines the unitary group G V. Given an o F -lattice in V, we define the conjugate-dual of as := {v V, h V (v, ) p F }. An o F -lattice sequence Λ in V is called conjugate-self-dual if there exists j Z such that Λ(k) = Λ(j k) for all k Z. As in [66], we may scale the indices of Λ such that (2.10) j = 1 and the period of Λ is always even. For example, if Λ is the lattice sequence representing {p k F } k Z in V = F, and h V is defined by (x, y) x c y resp. ϖx c y, then Λ is enumerated as (2.11), Λ( 1) = Λ(0) = o F, Λ(1) = Λ(2) = p F,, resp., Λ( 2) = Λ( 1) = p 1 F, Λ(0) = Λ(1) = o F, Λ(2) = Λ(3) = p F,. As in [15, Section 2.8], we define the direct sum of two lattices. Given F -vector spaces V i, for i = 1, 2, and o F -lattice sequences Λ i of V i of period e i, we define Λ = Λ 1 Λ 2 to be an o F -lattice sequence of V 1 V 2 of period e = lcm(e 1, e 2 ) by Λ(r) = Λ 1 (e 1 r/e) Λ 2 (e 2 r/e), for all r R. We can check that (Λ 1 Λ 2 ) Λ 3 = Λ 1 (Λ 2 Λ 3 ) and Λ 1 Λ 2 = Λ2 Λ 1, so that we can define the notion of direct sum inductively. Also, the sum of conjugate-self-dual lattice sequences is conjugate-self-dual. et (V, h V ) be a Hermitian space with an orthogonal decomposition V = i I V i with each V i isomorphic to E i as a Hermitian space as in the beginning of Section 2.3, and let D = D(G V, T, F ) be the set of torus embeddings. We will study the conjugate-self-dual lattice sequence Λ x for each x D, defined by the direct sum Λ i,x, i I where each Λ i,x is a lattice sequence in V i of period 2 defined similarly to (2.11) by Λ i,x (0) = o Ei Λ i,x (1) = Λ i,x (2) = p Ei if i I o ; Λ i,x (0) = Λ i,x (1) = o Ei Λ i,x (2) = p Ei if i I e. Hence Λ x is also conjugate-self-dual and of period 2.

15 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA Hereditary orders and compact subgroups. We continue with the Hermitian space (V, h V ) with an orthogonal decomposition V = i I V i. We also fix i I and define W = W i as in (2.5), equipped with a Hermitian form h W as in (2.6). In these two paragraphs, let G = G V or G W, and write G = G σ. For any lattice sequence Λ in V or W, we associate to Λ a hereditary order A Λ in à = End F (V ) or End F (W ) and its Jacobson radical P Λ. We then define, for r R, P r Λ = P r Λ and Pr+ Λ = P s Λ. s>r If Λ is conjugate-self-dual, then each P r Λ is conjugate-self-dual, i.e., it is σ-invariant. We define in A = Ãσ the sub-lattices P r Λ,F/F = P r Λ A and P r+ Λ,F/F = P r+ Λ A for all r R. We then define a decreasing filtration {UΛ r} r 0 of compact open subgroups of G by U Λ = UΛ 0 = A Λ, U Λ r = 1 + P r Λ, and U r+ Λ = 1 + Pr+ Λ, for r R >0. If Λ is conjugate-self-dual, which means that each UΛ r is conjugateself-dual, then we define UΛ,F/F r = UΛ r G and U r+ Λ,F/F = U r+ Λ G for all r R 0. We simply write U Λ,F/F = UΛ,F/F 0. (For a general connected reductive group over a p-adic field, there is an issue on whether a maximal compact subgroup is equal to its underlying parahoric subgroup. This issue does not concern us for unramified unitary groups: these two groups are equal in this case.) As in the notation of Section 2.3, we write E = i I E i and E = i I E i, so that E = T = T i I i and (E ) σ = T = i I T i where T i = U1 (E i /E i ). We also write o E = i I o Ei and U E = i I U Ei. For a fixed i I, we view T i as an elliptic maximal torus in G Vi. Given an embedding x D, we define a morphism I i,x : E G W by I i,x :E T i T M = G Vi G V G W, t = (t i ) i I (t i, t( c t 1 )) ( σi t i, I x (t( c t 1 )), t i ). We regard the image I i,x(e ) as a subtorus of the elliptic maximal torus T i T x in M. Suppose that Λ is a lattice sequence in W, with a decomposition i I Λ i where, for all r R, Λ i (r) = Λ(r) V i if i i and Λ i (r) = Λ(r) V ±i, where V ±i := V i, V i V i,+. When i I and x D are fixed, we call Λ an o E -lattice sequence if Λ i (r) is invariant by I i,x(u Ei ) for each i I, and write P r Λ,E the subset of elements in P r Λ centralized by I i,x(e ), for r R. Note that we write A Λ,E = P 0 Λ,E. For r R 0, we write U r Λ,E = U r Λ Z GW (I i,x(e )), which is 1 + P r r+ Λ,E, and similarly for UΛ,E. If Λ is furthermore conjugate-self-dual, we write UΛ,E/E r = UΛ r Z GW (I i,x(e )), U Λ,E/E = UΛ,E/E 0, and similarly for U r+ r+ Λ,E and UΛ,E/E.

16 16 GEO KAM-FAI TAM Finally, we recall the Cayley map (2.12) C : P 1 Λ U 1 Λ, X (1 + X/2)(1 X/2) 1. Its restriction on P n Λ defines a bijection onto U Λ n for all positive integer n, and is equivariant under the conjugation of U Λ. This defines an isomorphism P n Λ /Pn+ Λ = UΛ n n+ /UΛ. Moreover, the restriction of C on Pn Λ,F/F UΛ,F/F n is also bijective for all n, is invariant under U Λ,F/F, and induces an isomorphism P n Λ,F/F /P n+ Λ,F/F = UΛ,F/F n /U n+ Λ,F/F attices of higher ranks. Fix i I and x D, and recall that x corresponds to a partition I = I o I e. We define lattice sequences m i,x, M y i, and,x M z i in W = V,x i, V V i,+ as follows. Recall that we have fixed isomorphisms V i, = V i,+ = V i. et Λ i, and Λ i,+ be two lattice sequences in V i, and V i,+ respectively and both identified with Λ i,x in V i under the above isomorphisms. We define a conjugate-self-dual lattice sequence m ±i,x in V ±i := V i, V i V i,+ of period 6 by m ±i,x(r) = Λ r 1 i, ( 3 ) Λ r i,x( 3 ) Λ r + 1 i,+( 3 ), for all r R 0, and define Therefore, m i,x = m ±i,x i i Λ i,x. m i,x(r) = Λ r 1 i, ( 3 ) Λ x( r 3 ) Λ r + 1 i,+( 3 ), for all r R 0, and its period is 6. Following [66, Section 7.2.2], we define M y ±i and,x Mz ±i in V,x ±i, each of period 2, by assigning each of them to one of the following two lattice sequences: or 0 m ±i,x( 2) and 1 m ±i,x(3) 0 m ±i,x(0) and 1 m ±i,x(1) if i I o, 0 m ±i,x( 1) and 1 m ±i,x(2) if i I e, such that the reductive quotient U M w ±i,x,e/e /U 1 M w ±i,x,e/e We then define, for w {y, z}, U 3 (k Ei /k E i ) when w = y, U 2 (k Ei /k E i ) U 1 (k Ei /k E i ) when w = z. (2.13) M w i,x = M w ±i,x i i Λ i,x. is isomorphic to Again their periods are both 2. From now on, if i I and x D are fixed, we skip putting them in the subscripts of m or M w for notational convenience. Remark 2.4. Implicit in the construction above is that the decomposition W = V i, V V i,+ in (2.5) is exactly subordinate to a certain semi-simple stratum, in the sense of [66], with a suitable lattice sequence. We will recall the semi-simple stratum in our situation only in Section 3.3 because we don t need its full notion at

17 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA this moment. To check the exact subordination, we will recall the statements from [66]. With the notations in Section 5 loc. cit., we have W ( 1) = V i,, W (0) = V, W (1) = V i,+, V i = V i if i i, and V i = V i, V i V i,+ if i = i. (I) We first check the conditions [66, (i) and (ii) before Def. 5.1]. For (i), we verify that W (j) = i I W (j) V i for j { 1, 0, 1}. For j = 1, the statement holds since W ( 1) V i = 0 if i i and W ( 1) V i = W ( 1), which is V i, in our notation, if i = i. Similar statement holds for j = 1. For j = 0, note that W (0) is V and each W (0) V i is V i in our notation. For (ii), that W (j) V i is an E i -vector space for i I and j { 1, 0, 1}, is also easily verified since each V i can be identified with E i, and both V i, and V i,+ can be identified with V i and so with E i. (II) We then check that the decomposition (2.5) is subordinate to the stratum with lattice sequence m and M w (with w {y, z}) respectively. In the sense of [66, Def. 5.1(i)], this means that m(r) = (m(r) W ( 1) ) (m(r) W (0) ) (m(r) W (1) ) which is Λ i ((r 1)/3) Λ x (r/3) Λ i ((r + 1)/3) in our situation, so the statement holds just by construction. As a remark, a similar statement holds for M w, because when it is viewed as a function from Z to the set of lattices, its image is lying in that of m. (III) We then check that (2.5) is properly subordinate to the stratum with lattice sequence m, which means that it satisfies (II) and for all r Z and i I, there is at most one j { 1, 0, 1} such that (2.14) m(r) W (j) V i m(r + 1) W (j) V i in the sense of [66, Def. 5.1(ii)] (if there is no such j, then the left and right sides of (2.14) are equal). (a) Suppose that i i. If r = 3k, then (2.14) holds for j = 0. If r 3k, then there is no such a j. (b) Suppose that i = i, then (2.14) holds only when (r, j) is of the form (3k, 0), (6k + 1, 1) or (6k 1, 1) when i I o, and (6k + 2, 1) or (6k 2, 1) when i I e. For other r, there does not exist such a j. As a remark, it is clear that (2.5) is never properly subordinate to M w. For example, when i I o, w = y, and r = 1, (2.14) holds for all j. (IV) Finally, we check that (2.5) is exactly subordinate to the stratum with lattice sequence m and whose semi-simple element generates an F -algebra isomorphic to E = i I E i and normalizes m. Using the definition on [66, p.331], this means that it satisfies (III) above and the following conditions. (a) We require that a 0 (Λ (0) ) B (0) in loc. cit., which is A Λx,E = A Λx ZÃV (I x (E )) = o E

18 18 GEO KAM-FAI TAM in our notation, to be a maximal conjugate-self-dual o E -order in B (0) = ZÃV (I x (E )) = E, (b) For j = 1 or 1, with W (j) contained in V i, we require that a 0 (Λ (j) ) B (j) in loc. cit., which is A Λi,E = A Λi ZÃVi (E i ) = o Ei in our notation, is also a maximal o Ei -order in B (j) = ZÃVi (E i ) = E i. We will explain the reason of requiring the various stages of subordination in Remark 3.8 and Proposition Descriptions on filtration subgroups. For future computation, we provide the matrix descriptions of U k, for k Z and the lattice sequence m = m i,x or M w = M w i. In fact, we only list out U k,x for k = 1,..., e, because the others can be computed easily by periodicity. Recall that x D corresponds to a partition I = I o I e. Suppose that, for ɛ {o, e}, we denote V Iɛ = i Iɛ V i, such that each in has a corresponding decomposition Io Ie +. For J, K being, I o, I e or +, the notation P k in the (J, K)-entry of the matrices below represents Hom of ((m) K, (m + k) J ). When i I o, we have P 0 m = P 3 m = Also, and P 0 P 0 P 1 P 0 P 1 P 0 P 1 P 0 P 0 P 0 P 0 P 0 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 0 P 1 P 0 P 1 P 1 P 2 P 1 When i I e, P 0 m = P 3 m = m Z, P 1 m = P 0 M = y, P 4 m = P 0 M = z P 0 P 0 P 0 P 0 P 1 P 0 P 1 P 1 P 1 P 0 P 0 P 0 P 1 P 0 P 1 P 0 P 1 P 0 P 1 P 1 P 2 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 0 P 0 P 1 P 0 P 0 P 0 P 1 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 1 P 0 P 0 P 0 P 0 P 1 P 1 P 0 P 1 P 0 P 0 P 0 P 0 P 1 P 1 P 1 P 1 P 0, P 1 m = P 1 P 0 P 1 P 0 P 1 P 1 P 1 P 0 P 0 P 0 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 2 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 2 P 1, P 2 m =, P 1 M = y, P 1 M = z, P 4 m = P 1 P 0 P 0 P 0 P 1 P 1 P 1 P 1 P 1 P 0 P 1 P 0 P 1 P 0 P 1 P 1 P 1 P 0 P 1 P 1 P 2 P 1 P 2 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1, P 5 m = P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 0 P 0 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 1 P 0 P 0 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 2 P 1 P 1 P 1 P 2 P 1 P 1 P 1 P 1 P 1 P 2 P 1 P 2 P 1 P 1 P 0 P 1 P 0 P 1 P 1 P 1 P 0 P 1 P 0 P 1 P 0 P 2 P 1 P 2 P 1., P 2 m =, P 5 m =,. P 1 P 0 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 2 P 1 P 2 P 2 P 1 P 1 P 1 P 1 P 2 P 1 P 1 P 1,.,.

19 ENDOSCOPIC CASSIFICATION OF VERY CUSPIDA Also, and P 0 M = z P 0 M = y P 0 P 1 P 0 P 1 P 1 P 0 P 1 P 0 P 1 P 0 P 0 P 0 P 1 P 0 P 1 P 0 P 0 P 0 P 0 P 0 P 1 P 0 P 1 P 1 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0, P 1 M = z, P 1 M = y P 1 P 0 P 0 P 0 P 2 P 1 P 1 P 1 P 1 P 0 P 1 P 0 P 2 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 1 P 1 P 1 P 1 P 0 P 1 P 1 P 1 P 0 P 1 P Conjugate actions on unipotent subgroups. We denote V ɛ = V i,ɛ for ɛ {, +}, and the space n (j,k) = Hom F (V k, V j ) for j, k {, +} I. We denote by U and U the unipotent subgroups of the parabolic subgroup P and its opposite P respectively. They are the σ-fixed points of the unipotent subgroups Ũ and Ũ of the parabolic subgroups P and its opposite P respectively, where P is defined as the subgroup of G W stabilizing the same flag as in (2.7). The ie algebra of Ũ can be written as a decomposition of blocked root spaces n (,+) i I (n (,i) n (i,+) ),,. and similarly Ũ can be written as n (+, ) i I (n (i, ) n (+,i) ). Note that we used the same symbol σ for the involution (2.3) defined on the iealgebra ÃW of G W, then σ induces F -vector space isomorphisms respectively between the subspaces n (i, ) and n (+,i), n (,i) and n (i,+) for i I, n (j,k) and n (k,j) for j, k I and j k, which induce isomorphisms like (n (i, ) n (,i) ) σ = n (i, ), and so on. Also, n (+, ), n (,+), and n (j,j) for j I {, +} are σ-invariant, with fixed points space n σ (+, ), and so on, being the space of skew- Hermitian matrices u ni (F/F ). The torus T m = T i 0 T x in U m, where m = m i,x, acts on the ie algebra A W = Ãσ W by conjugation, leaving the blocks n (j,k), for j, k I {, +}, invariant. The induced actions on these blocks factor through suitable projections onto the components T i 0 and those of T x = i I T i,x. For example, the action of T m on n σ (+, ) factors through the conjugation of T i 0, and that on n (i, ) = Hom F (V, V i ) factors through left- and right-multiplications by the components T i 0 and T i,x on V and V i respectively. For being one of the lattice sequences m and M w = M w i,x, where w {y, z}, and for each m Z and j, k {, +} I, we denote We note that σ maps P m,(j,k) = {X n (j,k) X k (l) j (l + m) for all l Z}. P m,(i, ) onto Pm+ɛ,(+,i) and Pm,(,i) onto Pm+ɛ,(i,+), for i I, P m,(j,k) onto Pm,(k,j), for j, k I and j k,