T-Cell-Depleted Allogeneic Stem Cell Transplantation After Immunoablative Induction Chemotherapy and Reduced-Intensity Transplantation Conditioning in Treating Patients With Hematologic Malignancies

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following hematologic malignancies: * Acute myeloid leukemia (AML), meeting 1 of the following criteria: * In first complete remission (CR1), meeting 1 of the following criteria: * Adverse cytogenetics with minimal residual disease detectable by flow cytometry, cytogenetic analysis, fluorescence in situ hybridization (FISH), or polymerase chain reaction (PCR), defined as 1 of the following: * Complex karyotype \[≥ 3 abnormalities\] * inv(3) or t(3;3) * t(6;9) * t(6;11) * Monosomy 7 * Trisomy 8, alone or with an abnormality other than t(8;21), t(9;11), inv(16), or t(16;16) * t(11;19) (q23;p13.1) * Failed to achieve CR after primary induction chemotherapy * Secondary AML * In second or subsequent remission (CR2 or greater) * Acute lymphoblastic leukemia, meeting 1 of the following criteria: * In CR1, meeting 1 of the following criteria: * Adverse cytogenetics with minimal residual disease detectable by flow cytometry, cytogenetic analysis, FISH, or PCR, defined as the following: * Translocations involving 11q23, t(9;22), or bcr-abl rearrangement * Failed to achieve CR after primary induction chemotherapy * In CR2, if CR1 was \< 12 months * In CR3 or greater * Myelodysplastic syndromes (MDS) * INT-2 or high-risk by International Prognostic Scoring System * No MDS with Fanconi anemia * Chronic myelogenous leukemia (CML), meeting 1 of the following criteria: * Accelerated phase with treatment failure after imatinib mesylate * Blast phase * Myeloproliferative disorders, meeting 1 of the following criteria: * Agnogenic myeloid metaplasia with adverse-risk features, meeting at least 2 of the following criteria: * Hemoglobin \< 10 g/dL or \> 10g/dL if transfusion-dependent * WBC \< 4,000/mm\^3 OR \> 30,000/mm\^3 OR requires cytoreductive therapy to maintain WBC \< 30,000/mm\^3 * Abnormal cytogenetics, including +8, 12p- * Polycythemia vera or essential thrombocythemia in transformation to secondary AML * Myelodysplastic/myeloproliferative disease * Chronic myelomonocytic leukemia * Hodgkin's lymphoma or non-Hodgkin's lymphoma * Refractory lymphoma with progressive disease during combination chemotherapy * Relapse after OR ineligible for autologous stem cell transplantation (SCT) * Chronic lymphocytic leukemia * Treatment failure\* after fludarabine, chlorambucil, and at least 1 other salvage regimen * Prolymphocytic leukemia (PLL), meeting 1 of the following criteria: * T-PLL * Treatment failure\* after alemtuzumab and at least 1 other regimen * B-PLL * Treatment failure\* after fludarabine and at least 1 other salvage regimen * Multiple myeloma, meeting 1 of the following criteria: * Relapse after autologous SCT * Plasma cell leukemia * Adverse cytogenetics, defined as 1 of the following: * del(13q) = 11q translocation NOTE: \*Treatment failure is defined as relapse within 6 months OR failure to achieve remission * Less than 10% blasts in bone marrow and no circulating blasts in peripheral blood for the following diagnoses: * Primary or secondary leukemia * Refractory anemia with excess blasts * CML * Other eligible diagnosis in transformation to acute leukemia * Expected survival of approximately 1 year or less with conventional therapy * No active CNS involvement by malignancy\* * Prior CNS involvement with no current evidence of CNS malignancy allowed NOTE: \*Active CNS malignancy is defined by lymphoma: tumor mass on CT scan or leptomeningeal disease OR leukemia: blasts present on cerebrospinal fluid cytospin * Availability of a donor who is a sibling, parent, or offspring who shares 1 full haplotype (HLA-A, -B, or -DR) * Recipient and donor must have at least a 2-antigen disparity in either the host-versus-graft or graft-versus-host direction * Parent or offspring donor who is mismatched for a single HLA antigen (i.e., 5/6 HLA) is allowed * No sibling donor who is 6/6 HLA-matched OR mismatched for a single HLA antigen (i.e., 5/6 HLA) * No unrelated donor identified in a prior or current National Marrow Donor Program registry search PATIENT CHARACTERISTICS: Age * 18 to 55 Performance status * ECOG 0-2 OR * Karnofsky 60-100% Life expectancy * At least 3 months Hematopoietic * See Disease Characteristics * Absolute neutrophil count ≥ 1,000/mm\^3\* * Platelet count ≥ 20,0000/mm\^3\* (without transfusion) NOTE: \*Lower values may be accepted at the discretion of the principal investigator or study chairperson if due to bone marrow involvement by malignancy Hepatic * ALT and AST ≤ 2.5 times upper limit of normal (ULN)\* * Bilirubin ≤ 2.5 times ULN\* * Unconjugated hyperbilirubinemia consistent with Gilbert's syndrome allowed * No chronic active hepatitis B infection * Hepatitis B core antibody positive allowed provided patient is surface antigen negative and has no evidence of active infection * No hepatitis C viral infection * Seronegative for anti-hepatitis C antibody and detectable hepatitis C viral RNA by reverse transcriptase-polymerase chain reaction assay NOTE: \*Higher levels may be accepted at the discretion of the principle investigator or study chairperson if such elevations are due to liver involvement by malignancy Renal * Creatinine ≤ 1.5 mg/dL OR * Creatinine clearance ≥ 50 mL/min Cardiovascular * LVEF ≥ 45% Pulmonary * DLCO ≥ 50% of expected value (corrected for blood hemoglobin level and alveolar volume) Other * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for 1 year after study participation * HIV negative * No active infection not responding to antimicrobial therapy * No psychiatric disorder that would preclude study compliance or informed consent PRIOR CONCURRENT THERAPY: Biologic therapy * See Disease Characteristics * At least 2 weeks since prior monoclonal antibody therapy Chemotherapy * See Disease Characteristics * At least 2 weeks since prior systemic chemotherapy Endocrine therapy * Not specified Radiotherapy * Not specified Surgery * Not specified Other * Recovered from all prior therapy * No administration of tyrosine kinase (TK) inhibitors, including imatinib mesylate and dasatinib, during the conditioning regimen; TK inhibitor administration may resume 28 days after transplantation