Luigi Mangione's family appears to be distancing themselves from the Brian Thompson murder investigation ... in which the Ivy League graduate is the main suspect.

Attorney David B. Irwin, who has been advising the Mangione family following Luigi's arrest, tells TMZ ... the suspected shooter's loved ones are in the dark about the case -- saying they don't "know all the facts."

In fact, the attorney tells us the family only knows what they "read in the papers" ... making it clear they aren't involved in Luigi's legal woes.

They did have a short, albeit sweet, message for Luigi ... who the attorney says the family "loves."

He adds ... "We wish him the best."

The show of support from the Mangione family comes as Luigi faces murder and gun possession charges in the fatal shooting of UnitedHealthcare CEO Brian Thompson.

As TMZ previously reported, Luigi was first named as a person of interest earlier this week after Thompson was gunned down in front of a midtown Manhattan hotel on Dec. 4.

Luigi was picked up in an Altoona, PA McDonald's after a customer spotted Luigi and recognized him from widely shared suspect pics ... prompting an employee to call the authorities.

From there, Luigi was arrested on firearms and forgery charges in Pennsylvania before being hit with a second-degree murder charge by the Manhattan D.A.

He's currently fighting extradition to NYC ... with his attorney previously sharing Luigi plans to plead not guilty.